"μου αρέσει το βιβλίο" is similar to "das Buch gefahlt mir" , where "mir" is dative. (in Romanian we are using dative, too : îmi place cartea ; îmi = mir , dative) . To me it looks like μου/σου are genitive pronouns that play the role of an ancient dative. My guess is that at some point in history the dative and the genitive were merged together , so that currently in modern Greek we find only the genitive (playing both roles in different contexts).

unfortunately, I don't speak German so I can't make the connection between the Greek phrase and the German ones you wrote. But I know Greek grammar! So, you're right in "μου αρέσει" , "μου" is genitive. It's not an old genitive form, it is just the genitive form of the weak personal pronoun. This is the most common way to say "I like" in Greek. If we wanted to translate it more literally in English, it would be something like "this is likeable to me". However, I can't express the genitive form precisely in English. The verb "αρέσω" can also be used with the accusative like this: Αυτό αρέσει σε εμένα ==> this is likeable to me (again the same translation). Here, "εμένα" is the accusative of the strong personal pronoun ( εγώ ). "σε" is a preposition, like English "to". Having said that, it's more common to hear (or read) "αυτό μου αρέσει" than "αυτό αρέσει σε εμένα" in Greek. So I would say that this verb is used more with the genitive than the accisative form of the personal pronoun.

there is no something exact translation of that verb in english In english you can say: i like..... in german language is much easier,germans said : ich mochte ...ich mag