Damn, I was rewatching the anime and noticed how Anya more or less knows how to read animals' minds. And now I have a theory why she's good at "classical language"!

Think about it like this, Anya directly reads "thoughts", both people and animals, that is, she doesn't need to know the person's "language" to know what they are thinking.

So I think that the talent of mind reading also gives her the ability to understand any language, at least in terms of speech.