I'm a bit confused. As far as I know realism is opposed to idealism.
But we can say that Plato was an idealist when speaking about Plato's forms. We also talk about Plato's realism in the philosophy of mathematics for instance.
I also read that Kurt Gödel was a realist who believed on the objective existence of mathematical objects (Source : https://plato.stanford.edu/entries/goedel/#GodRea). But aren't these ideas related to what we call idealism ?
Both Plato and Gödel were mathematical platonists. Both held that mathematical objects existed abstractly and outside of spacetime. This is what we would call mathematical realism. This position is different from just the Forms because even Plato in The Republic and other dialogues distinguishes between the type of being exhibited by the Forms and by the mathematical objects respectively. Nevertheless the being that both Plato and Gödel take mathematical objects to have is what we call platonism, though emphasize more of how they exist just outside of spacetime than that they're 'ideal'; use the word 'abstract' instead.
The main thing happening here is a shift in the meaning of terms. Specifically, the word "realist" has had many many different uses over time. The basic idea is that a realist believes something is real, i.e. that such entities do in fact exist, that they are metaphysical objects in their own right (rather than existing as modalities on the mind).
Realism when used in reference to mathematical realism and Plato is that X is believes Y do in fact exist.
So Plato is a realist about Platonic Forms. Mathematical realists believe numbers do in fact exist.
Plato's view stands in contrast to Aristotle's view -- which while also realist with respect to forms does not think the forms exist as ideas. Instead, they exist as essences in substances. In contemporary language, this is the debate over universals. Anti-realist views think that these things only exist in our heads as words or concepts. (Realist views can of course accept that we also make words or terms for these things).
The term idealism doesn't generally occur in this debate. Idealism instead refers to views where things only exist due to the mind. It's a partial offshoot of rationalism generally associated with Schopenhauer. Hegel is often called an idealist as well, but I would argue this is a misnomer.
