Both the dative of possession and a possessive adjective could be used here, actually.
In Allen and Greenough's Latin Grammar, it states:
The genitive or a possessive with esse emphasizes the possessor; the dative, the fact of possession.
It's very similar to English.
I can ask in Latin "Estne liber tuus (aut alterius)?", which in English is "Is this your book (or someone else's)?". Here, I'm not questioning whether or not it's possessed. I'm questioning who is possessing it.
Conversely I can ask "Estne liber tibi (aut non)?" or literally in English "Is this book for you (or not)?" (though more naturally we might ask in English "Does this book belong to you?" or "Do you own this book?"). Here, I'm not asking about the possessor, since I only give one possible possessor. The question is whether or not it's possessed by you.