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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 it's legislature (This is actually unique. Nebraska is also the only state with a Unimerical 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 Alexia Ocosio-Cortez (Spelling appologies if they were made) 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 ...