I'm curious if any parties running for student government, i.e. student unions, ever run as Democrats, Republicans, Libertarians, Green, Socialist, etc?
I am not aware of any such student government in the United States (and in the course of serving in student governments at two different institutions, I encountered a great many of them in student government to student government dealings).
Many colleges and universities have clubs associate with political parties, which typically focus on partisan politics in their locality and state and in the nation, rather than on campus politics per se.
At least one college of which I am aware (Oberlin) had (for at least some period of time and may still have) some seats in the student government that were elected from student club nominees by club representatives (thus there might, for example, be a single Republican club representative in a 30 person student senate elected by leaders of other clubs), but I am not aware of any student government whose primary factions correspond to the national political parties.
Longer Answer: Why Might This Be The Case?
There are multiple reasons why this is the case:
Student government is often less divided by faction than governmental political parties because they represent the student body vis-a-vis other factions within the university (faculty and the administration) which makes it valuable to be able to present a united front.
Student government often has little policy making authority other than to spend student activities funds and is often bound strongly by tradition in that function (i.e. not all student governments have a say in college or university policy making, although some do, similarly not all faculty governance institutions have a say in college or university policy making although some do).
To some extent this makes sense because the issues addressed in student government often don't map very well to the issues addressed by national political parties. There isn't a clear Republican or Democratic stance on whether State U.'s math department needs another professor, or whether sorority rush should be held in the spring or the fall, or what theme should be chosen for the homecoming dance.
To some extent this fits with a U.S. tradition, at least since the progressive era (late 1800s, early 1900s) of formally non-partisan municipal, school board, and special district government in most states. While many countries have close vertical integration of political parties at all levels of government, the United States does not. There is modest coordination between federal and state major political parties, and modest coordination between state major political party officials and local major political party officials in the handful of partisan local offices (usually at the county level an township level), the nature of county political offices (e.g. coroner, surveyor, treasurer) which is often ministerial in nature leaves little room for policy coordination with local officials even when they are partisan. Township trustees, for example, in most states that have a township level of local government, usually receive property tax funds in a property tax system administered mostly by the county and use those funds to construct and maintain local roads and bridges, and despite often being elected on a partisan basis, there is no Republican or Democratic way to fill a pothole and these bodies cannot usually raise funds without the approval of the voters or another governmental body.
To some extent this makes sense because in most colleges and universities, one party or the other would have a permanent majority. Brigham Young U. and Liberty U. would always be controlled by Republicans. Oberlin College and Evergreen U. would always be controlled by Democrats. As in the case of non-partisan local governments, lack of affiliation makes it possible to members of a party that is a clear minority in the locality have a viable shot at winning office in local government on a non-partisan platform.
Yet another reason is that many student governments are formally constituted either as non-profit entities under 501(c)(3) of the Internal Revenue Code, or as divisions of non-profit colleges and universities under 501(c)(3). Under the Johnson Amendment, 501(c)(3)s are not allowed to endorse political candidates or engage in partisan politics and a student government organized on a partisan basis coinciding with state and national political parties could jeopardize the 501(c)(3) status of either the student government itself (if organized as a separate entity) or the college or university as well (if organized as a division of the college or university rather than as a separate entity).
Political parties exist partially to intermediate between voters and candidates, but this is much less important when the number of voters per candidate is very low as is often the case in student government. A related observation is that college or university specific identified party-like factions in student government tend to be more common in universities like the U. of Florida or U. of Colorado where the voter to candidate ratios tend to be higher. The largest universities in the U.S. have about 50,000 students at a given campus and often a dozen or more student government officials who are elected, which is still a very low ratio compared to most general governments outside the college or university context, and the average is much, much smaller.
Refraining from a formal affiliation with national political parties discourages national political parties from attempting to interfere with or supervise student government groups and insulated these parties from being embarrassed by these student government groups to their detriment in governmental elections beyond a campus.
As an aside, the predominant means by which U.S. college and university student governments are selected is some form of election by the student body.
Many U.S. colleges and universities also have two parallel systems of student government - one that includes all students and another "Pan-Hellenic Council" that consists of representatives from each of the fraternities and sororities on campus. Colleges and universities with cooperative housing and/or dining systems also often have an umbrella organization that represents all of the individual cooperating housing and cooperative dining locations on campus. On some issues these parallel systems of representing students present a united front for students vis-a-vis the faculty and administration, while on other issues these parallel representatives of the student body may be divided.
Yet another parallel form of student government often exists on U.S. colleges and universities within particular academic departments with majors in the department collectively, or through representatives, having some voice (and sometimes even a vote) on decision making within the department separate and apart from campus level student governance organizations.
A few universities (e.g. University of Michigan-Ann Arbor) have predominantly student organizations that aren't strictly student governments, but are mostly made up of students represented collectively (such as the Ann Arbor Tenants Union).
Finally, graduate students at a number of universities (typically where graduate students have teaching and/or research responsibilities) are sometimes unionized separate and apart from a traditional student government.
Student governments in Japan, in contrast, historically were either self-perpetuating, or are selected by the administration often based upon academic performance in a system somewhat analogous to the "prefect" system in British non-governmental independent schools. To make an American comparison, student government officials in Japan are partially a bit like R.A.'s (resident assistants) in the dorms of U.S. colleges and universities (i.e. students given authority over other students by the administration). This parallels the fact that many Japanese companies have "company unions" of employees sponsored by the companies themselves rather than in an adversarial relationship with the company's management (something prohibited by U.S. labor laws).
Also, while student governments are almost never organized on a partisan basis, some states (e.g. Colorado) have college and/or university systems whose governing board is elected on a partisan basis, and more have governing boards whose members are appointed by politicians in a manner such that the appointing politician always selects a partisan loyalist for the post.