I translated this as "His girlfriend likes philosophy." First question, how would this be expressed in modern Greek? Second question, is there a syntactical rule that when a noun is used as the indirect object of "αρέσει", is it necessary to repeat the corresponding pronoun?
Would it ever be correct to say the phrase this way: «Το κορίτσι αρέσει η φιλοσοφία.»
You can't have two subjects in one sentence, and the verb αρέσω doesn't take a direct object, so you can't have one subject and one direct object with that verb, either. The sentence is simply wrong.
There is no dative any more in Greek so genitive is used instead
του κοριτσιού( gen. of το κορίτσι) του (an unnecessary repetion of the gen of the girl's definite article το) αρέσει η φιλοσοφία/ to the girl, the( rep. of the first the) the philosophy is pleasent e.g. the girl likes philosophy