Yes, Russian works in a way that a name has a common "short form", and for Anastasia it's "Nastya", while for "Anna" it's "Anya". So an "Anya's" full name would more commonly be "Anna", and that's what most Russian people would assume the girl's name is. However, sometimes people choose individual nicknames, so there would be nothing impossible for a particular Anastasia to ask to call her "Anya" in short if that's her choice.
I'd assume the common shortenings for "Anastasia" would be, in order of commonality: Nastya -> Nastasya -> Stasya (I had a classmate who went by this, even!), and only then "Anya".
I've even met a few Anastasia's annoyed foreigners call them "Anya's", but I think that also takes a particular kind of person.
(And in Japan, some don't want to be called "eggplant tea" ナス社茶)

As for FGO and the Grand Duchess Anastasia, I think they watched that 1997 cartoon too much, it makes the same mistake. In letters, she was shortened either as "Nastya" or "Nastas'ka" (another "small"-showing suffix "-ka" is added here).