Is the sine function periodic in $x$ or $y$?

Yes your function is periodic on the interval $2pi$.

Don't get hung up on these details of language. One might argue that because $x$ is the independent variable, it moves in arbitrary ways (can be chosen arbitrarily) and so it would make sense to say "the sine function is periodic in $y$". But one could also say "the sine function is a periodic function of $x$" (because it is a function of $x$).

In the end, whatever you say, people will understand what you're trying to say: "the sine function is a periodic function" omits all details and yet, everyone understand what you are trying to say.