Both shows lean heavily on the underlying message that we can all be better people, and that we all benefit from cooperation, and that these two things are interrelated: We are better people for working together, and working together is enabled by being better people. There's also a strong implication in both that learning and discovery are inherently good things, a part of becoming a better person. It's almost like Community is Star Trek set in the early 21st century, presumably in between the Eugenics Wars of the 1990s and the mid-century Third World War.
The show also follows basically the same format, with a small, diverse ensemble cast nominally centered around a charismatic leader figure, but with most story attention actually going to the more interesting other characters (the emotionless outsider-looking-in being a special fan-favourite). Each episode has some external drama to resolve, but the real enduring focus of the show is the friendships of the core ensemble.
While they supposedly have a very large home structure to roam around (in addition to all the external places they can visit on an episodic basis), we mostly only see their central gathering room (where everyone sits in a circle and the leader has a special seat apart from the rest), a communal eating/social area, a few stock corridors and little else. Community even goes so far as to introduce the holodeck (in the form of the dreamatorium) which may be solely imagination powered, but that's no less of a stretched shoe-horning than some of the anachronisms crammed into ST:Enterprise.
The multiversal nature of Community fits well with established Star Trek canon, and I think there's little doubt that while the world of Community may be closely modelled on our own early 21st century, it's still a science fiction series in disguise.
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For purists of both sorts, I'm not insisting that Community should be considered Trek canon - the LeVar Burton issue alone would be a nightmare. But it does clearly fit a very similar social and entertainment purpose, and I love both in very much the same way. We all need many more shows in this vein.