"I like the shirt."
Translation:Μου αρέσει το πουκάμισο.
I do not think that accent should be used since in Το πουκάμισο μου αρέσει, ΜΟΥ must belong to "αρέσει". Otherwise, if "moυ" is possessive we would write: **Το πουκάμισό μου". (Futhermore with the possessive "μου" we cannot continue with "αρέσει" without another "μου": Το πουκάμισό μου μου αρέσει)
You are right it's not needed because of the double accent thing. Truth is, the monotonic system has some problems, and there is no official answer to the above matter (whether to add the accent on μου because there would be a double accent on πουκάμισο otherwise). I say it's good to accent the pronoun in every situation like that, because not only it follows the verbal speech (which actually accents the pronoun) but also because otherwise it's a constant quiz (Do I have to add the accent? Oh no the word is accented in the antepenult, so there would be a double accent). And I am for accenting two μου that come one after the other (Ο πατέρας μου μού είπε=My father told me), not only because it follows the rules of verbal speech, but it's aesthetically efficient. But this is my opinion, and there is no official answer to that.
With the possessive, we can continue without another μου. Το πουκάμισό μου αρέσει=People like my shirt/my shirt is liked. Αρέσω is an active verb in Greek and can stand as is.
Oh. Μου is genitive of εγώ. Αρέσω has indirect object which is genitive nowadays since Greek does not have dative any more. One complication with αρέσω is that the En. subject becomes the Gr object and vv. One can use "please" to get it right: I like the shirt - The shirt pleases me- Μου αρέσει το πουκάμισο
You are on the right track with dative. The problem is that Modern Greek does not have dative so they use genitive instead
I saw that you read Spanish. In Greek αρέσω is similar to gustar in Spanish but with genitive instead of the non-existing dative. The same goes for γουστάρω but it is a slang word borrowed from Italian.