Learn Finnish: Se kuppi on hänen.

The word hän means he or she. By adding the genitive (possessive) ending -n, it becomes hänen, which means his or hers. This shows that the cup belongs to them.

How do I know if hänen means his or hers in this sentence?

You can only tell from the context! Finnish does not use gendered pronouns. hän is used for both he and she, so the possessive form hänen stands for both his and hers.

Doesn't se usually mean it? Why is it translated as that here?

You are right that se usually means it. However, it is also used as a pointing word to mean that. When you put it right before a noun, like se kuppi, it translates to that cup.

Could I say se on hänen kuppi instead?

Yes, but the meaning changes slightly. se on hänen kuppi means it is his/her cup. Our sentence, se kuppi on hänen, means that cup is his/hers. Both are correct ways to use the possessive hänen.