Word to describe a mathematical variable that repeats, like an angle or time

You could use periodic if the repetition follows any kind of function.

You might say that the variable is cyclic although that term more correctly applies to the values the variable can have than the variable itself.

Instead of talking about the variable, I would say that the operations on that variable use modular arithmetic.

What you've described is referred to as "wrapping" and a variable that wraps around is called a wrapped variable.

As jimreed rightly points out, the cause of wrapping is due to modular arithmetic and it is very uncommon to explicitly mention that a variable is "wrapped" because it is assumed to be understood. The only time I've seen mathematical/scientific articles use the term wrapped variable is when they also talk about "unwrapping" the variable. So in your clock example, you would add 12 after each time it completes a full circle to get an unwrapped variable.

I don't have a wiki link to the definition (because, as I said, it's not used commonly). However, the article on wrapped distribution uses the term wrapped variable.

This is a math question.

The answer is yes, periodicity isn't confined to being in one dimension.

Euler's formula sums up the idea of the sum e^{jt} = cos(t) + jsin(t) travelling in a circle in the complex plane as t goes from 0 to 2pi. See also phasors.