Understanding the phrase "belly matches paws"

Solution 1:

This is not a fixed idiom, but the comedian's invention.

You have understood the literal sense of the line correctly: the speaking cat admires the fact that the other cat's paws and belly are the same color.

The joke (such as it is) lies in the context. The comedian is talking about these two indoor cats as "prison gays", environmentally driven to homosexuality. An enduring stereotype about homosexual men is that they are especially discerning in matters of fashionable dress. Consequently the speaking cat's expression of admiration for the other cat's ensemble is an indirect but transparent announcement of his homosexual attraction.

Ha.ha.

Solution 2:

I'd add that the formulation is probably influenced by the crude cliche "the carpet matches the drapes," alluding to a woman whose hairdo (which hangs long like window draperies) has not been dyed, and so is the same color as her pubic hair (which is short and low like a carpet).