And is the two-party system currently in place simply an emergent consequence of human nature?

We often find that there exist sets of politically controversial claims that are strongly correlated, even though the validity of each claim is entirely orthogonal. For example, the normative claim that abortion is morally acceptable, and the descriptive claim that humans are responsible for climate change, are entirely independent of one another, yet we find that almost everybody is either a proponent of, or a detractor of both claims. Certainly more than we should expect if acceptance of either claim is a statistically independent process.

Is it possible for there to be a government, whose laws and constitution are identical to that of the US, in which each senator and member of congress is essentially their own "party" with their own unique set of positions on issues, and where the correlation between different members' position on issues only reflects the actual correlation between the validity claims of the issues themselves, and not, say, some other latent construct like a "party"? And that the president was just some other candidate that had no common affiliation with any senator or member of congress? Is the current partisan system purely a consequence of human nature? Or is partisanship hard-wired in US law?

George Washington said:

However [political parties] may now and then answer popular ends, they are likely in the course of time and things, to become potent engines, by which cunning, ambitious, and unprincipled men will be enabled to subvert the power of the people and to usurp for themselves the reins of government, destroying afterwards the very engines which have lifted them to unjust dominion.

Washington opposed the development of political parties. The design was that the country would not have parties. But people banded together and made parties, thwarting his design.

There is nothing in the United States constitution requiring parties. There is quite a bit of law promoting them though. For example, it is much easier to get on the fifty-one ballots (fifty states plus Washington, DC) for people who win the Democratic or Republican primaries.

The one constitutional encouragement for the two party system is the national nature of the presidential election. In a parliamentary system, it is easier to have multiple parties. The first-past-the-post system also pushes towards two parties per district but is not constitutionally required. It could be changed legally. Duverger's law.

A different system could be implemented, but it is unlikely that the existing parties will do so unless they determine that the system itself is making it hard for them. For example, moderate Democrats and Republicans might band together to make a system that allowed moderates to compete better. In many states, primaries are limited to just members of the party. So moderates like Joe Manchin and John Kasich are limited to just the moderates in their own party, while many of their natural supporters are of the other party or are independents.

The current partisan system is not constitutionally mandated, but it is legally self-perpetuating. The parties resist reforms, like non-geographic districts, that would produce better representation but reduce their own power.

U.S. Political Parties are not hard wired into U.S. Law. In fact, Nebraska explicitly has a non-partisan rule in its legislature (This is actually unique. Nebraska is also the only state with a unicameral legislature). United States Constitutional Law requires all States to be "Republic in Nature" so presumably a representative democracy with no monarchy will suffice that demand. All 50 states qualify as semi-direct democracies to one degree or another.

Nebraska primaries are different in this nature. Since there are no parties given, primaries are open to all citizens in the district who vote for the general candidates from a list. The top two best performing candidates are pitted against each other in their district's general election. Since the U.S. protects the right to free association, Nebraska cannot stop candidates from joining political parties, so most Senators are actually in a party dynamic... however, generals could result in Democrat vs. Democrat or Republican vs. Republican or Republican vs. Democrat, depending on the party affiliation of the top two primary candidates.

It also needs to be pointed out that in the United States, the political party system is a weak party system. This means that while Republicans or Democrats in party leadership may want their party members to vote a certain way, they can't force them to. During the most recent shut down, Representative Alexandria Ocasio-Cortez famously broke ranks with the Democrats and voted against the spending bill with Republicans (though not for the same reasons as the Republicans. AOC refused to vote for the bill because she was opposed to Funding ICE. The Republicans voted against it because it didn't fund the border wall).

Another recent famous aisle crossing was Senator John McCain's voting against a Republican bill that would have made sweeping changes, defeating it by a single vote.

One of the deepest darkest secrets of the U.S. Legislature is that... individual Democrats and Republicans are actually on very friendly terms with each other. Most do have friends that are in the opposite party... in fact, committee chairs are usually get along with their counter part Ranking Members as most committee leaders are long serving members... they do a lot of behind the scenes work with each other. And some congress men and women do have an unspoken dislike of other members of their own party (Nancy Pelosi won Speaker on very narrow margins... and she was unopposed). In fact, the most important room for making deals with the other side isn't the floor or the office of a party leadership... but the congressional gym. It's one of the few areas of the congressional offices where only congress members (and staff who keep up the facilities) are allowed to enter (so the staffers in their offices and the news cameras can't report on the details). A crafty Democrat can negotiate with a Republican while he spots the Republican's bench press.

I think a look at global trends shows that more than two parties are an emergent property of human nature, as long as the political system lets them get away with forming wings within parties and splinter groups within factions. The answer by Brythan quotes Duverger's Law, which shows that a two-party system emerges under special conditions.

Having a formal political party merely gives legal structure to the human tendency to form groups. Government without parties may be possible at a small town level, where everybody knows everybody and the issues are local, but imagine the mess if every budget proposal had to be negotiated by 435 people around a big table, each with an equal voice in the outcome.

The concept of committees also relies on party affilation, or everyone would want to be on Appropriations and few would take Indian Affairs ...

A lot of people are commenting about human nature and its role in partisan systems forming. As a matter of interest, I think we can expand on this by comparing the Westminster system used in the UK and several Commonwealth countries.

For specific examples, I'll be using Australia to illustrate, as (a) it has a single written Constitution, unlike the UK, and (b) as an Aussie it's the one I know best. But I think that what I've said below is also true of the UK, and probably mostly true of other Westminster democracies.

In summary, political parties are mostly absent¹ from the written law, but are a key element of convention. And convention is perhaps equally important as written law in how a Westminster government works.

The Executive and Legislature

Notionally, the executive and the legislature are separate bodies, although by law and/or convention², all members of the executive (aside from the head of state) are also members of the legislature. And notionally, it's the head of state who appoints the other members of the executive (the ministers).

In practice, the head of state first picks the head of government (the Prime Minister), and always chooses the leader of the party that has a majority in the legislature's lower house. The other members of the executive are whoever the Prime Minister tells the head of state to appoint³.

Consider what happens when no party has a majority in the lower house. The head of state decides which party is more likely to consistently get its votes passed by cobbling together enough support from outside the party. That party's leader becomes PM. If that shaky ability to win votes doesn't last, the legislature gets dissolved and an election is held.

So to run the government, you have to have the reliable support of a majority of legislators. That practically requires that parties exist, and that party members consistently vote the same way.

Election Systems

Election systems that favour major parties result in more stable government, and so are more common, even if they're arguably less democratic. More specifically, since it's a majority in the lower house that you need, the lower house's election system is almost always one that favours major parties. (In the UK, it's first-past-the-post; in Australia, it's single-candidate preferential.)

The upper house (where it hasn't been done away with) is expected to be more diverse and can have completely different election system. (The UK's House of Lords is unelected, while Australia's Senate is elected on proportional representation.)

"The Government"

The notion of power being tied to a party is so ingrained that the currently-dominant party is called "the Government" for short. Technically you could say that this refers to the executive, but in practice it also includes non-minister members of the legislature.

Further reinforcing not just a party system, but a two-party system, is the role of the Opposition (entirely a matter of convention). While the dominant party is the Government, the runner-up is the Opposition. Their job is to scrutinise the Government's actions, and present an alternative set of policies for the voters come the next election, because they (unlike minor parties) have a realistic chance of being the next Government.

Even the very layout of the legislative chambers is based on a two-party system. The Government sits on one side, the Opposition on the other, and minor parties sit on the "cross-benches".

Public Perception

The voting public knows and accepts the conventions, probably better than the written law. They almost universally choose a party to vote for (or against), not a candidate.

An interesting situation (and very topical in Australia) is a change in party leadership. When the reigning party's leader loses that leadership due to internal issues, that naturally results in them also being removed as Prime Minister. This tends to annoy the voters, who see themselves as "voting for a Prime Minister" at election time⁵, even though technically only one electorate directly voted for the PM (who is also their local member).

What If...?

You could imagine a non-partisan government, where the head of state actually uses the power that the written law gives them (but which convention says they should never use) and picks whoever they damn well please to be in the executive, regardless of party. But it just doesn't happen⁴; it'd be seen as a flagrant breach of convention and a slide towards dictatorship.

¹ Originally, the Australian Constitution didn't mention parties at all. A later amendment added a requirement that, when a senator gets replaced, their replacement has to be from the same party.

² The Australian Constitution allows non-members of Parliament to become ministers only for a limited time. In practice, it just doesn't happen.

³ This decision may be made at the Prime Minister's own discretion or by the party caucus.

⁴ A unity government is kind of close, but not really.

⁵ Or at least, the media can always sell the story that the people's chosen PM was kicked out by undemocratic intra-party squabbling, not an election.