Tell me who you vote for and I'll tell you how X you are.

It's not an argument; it is a methodology based on opinion. What the person is likely doing is using black and white thinking to portray the candidate as a one-dimensional character, that is, taking a single position of theirs or personality trait, exaggerating it, and defining them by it. Then, the person voting is being judged on their implicit support of that one position or personality trait while ignoring all the others. There is a unjustified assumption that the voter agrees with the candidate on their position or admires the candidate for their personality trait that makes them X.

It is possible that the voter has provided a reason for their vote that justifies them being X, but the statement as you have written it does not indicate that, so it is problematic but not fallacious.

It's context-dependent.

People vote in-line with their political views. So if a person is homophobic, they will feel validated by a homophobic candidate - and are thus more likely to vote for them. If Gary is running for public office, and goes on and on about how he dislikes gay people, then he will attract homophobes as his support base. Thus, if Laura votes for Gary, she is likely to be a homophobe herself (especially if Gary is very explicit and condemning of homosexuality, that there is no ambiguity in what he is saying).

On the other hand, this relies on society moving forward on LGBT rights a little bit. Assuming homophobia is the norm, all candidates may be homophobic. In this case, there isn't a clearly preferable option. Thus, voting for a homophobe indicates less about the person voting than in the first example.

The statement in the OP is not an argument; more like a heuristic to determine whether someone is bigoted or not. Sometimes it works, and sometimes it doesn't. And this is without getting into the messy argument over definitions of homophobia. However, it can be called a non sequitur in the second example.