This question may be too naive, in which case I apologise in
advance. Anyway, it is a well-known fact (see e.g. Milne's notes)
that any abelian variety A has only finitely many direct factors
up to automorphisms of A. (Here a *direct factor* of A is an
abelian subvariety B for which there exists another abelian
subvariety C of A such that $A \cong B \times C$.)

My question is: how much is known about the corresponding
question for *arbitrary* abelian subvarieties, rather than
direct factors? That is, is it known whether every abelian
variety A has finitely many abelian subvarieties, up to
automorphisms of A? If not, what's the best known result in this
direction?

I've asked a couple of people about this, and their opinion seems to be that it's "more or less" known. But I would like something a little more concrete, if possible. Any relevant references would be appreciated!