Given what the political/cultural situation is supposed to be in the Star Trek universe, I would say that Republicans as we describe them today no longer exist in that timeframe.
If pressed, I'd say that Kirk is a Communist, in the ideal, positive sense.
However, I think that Kirk's politics are irrelevant. He's the product of a successful society, and therefore probably embodies whatever values, institutions, and policies have made that society successful. But what we tune in for is his personality, qualities as an individual.
The politics of his society have given him the freedom to focus on things
other than politics. And that's what he does. He's too busy exploring deep space, conducting diplomatic relations with inscrutable aliens and godlike beings, boning space women and being a linchpin in the Federation's far-flung military presence, to bother with politics.
So yeah, I'd say he's a Communist, because as far as I can tell, the Federation is closer to Ideal Communism than anything else, and of course a successful member of a successful Communist society is going to be Communist themselves. But I don't think it's really relevant to the story that's being told about Kirk.
Also, I hate it when Politicians try to co-opt a fandom to their cause.