Hitherto Hogwarts has been the place where Harry escapes to from the prison of the foul little house on Privet Drive. Now it becomes a castle under siege, guarded by creatures even ghastlier than the Dursleys. It is no longer the locus of fun. The locus of fun has shifted outside the walls- to the holiday village of Hogsmeade and- guess what- Harry is forbidden to go there.
Then there's that other prison: the Guantanamo Bay-like hell-hole of Azkaban. In the first two books evil has been concentrated in the person of Voldemort (the enemy, the other, the dark lord, out there) here Voldemort never appears except in discourse and the focus of evil is Azkaban and its disgusting Dementors who- disturbingly- are on our side.
So, if we employ Dementors and send people who break our rules- like the innocent Hagrid- to a place where it's guaranteed they'll lose their minds if not their very souls- how exactly are we better than Voldemort? Dumbledore refuses to let the Dementors into the castle grounds, but they come anyway. Hogwart's defenders are weaker than we formerly imagined. Not only is Dumbledore unable to keep the Dementors out, he is shown to be answerable to the weak, foolish and arguably corrupt politician, Cornelius Fudge, who, in his turn, is under the thumb of Voldemort's old ally, Lucius Malfoy.
So who can we trust? None of the authority figures is to be relied on. Hagrid is a fellow victim, Lupin is an unknown, Snape is unreadable. And Dumbledore, splendid as he may be, has his hands tied. In the final crisis, he sends two children out, unbacked and unprotected, to do the work he cannot- dare not- do himself.
So this is escapist fiction, is it? Well it's still fun, there are still lots of jokes (I love Peeves; why can't Peeves be in the movies?) but this parallel universe of unseen enemies, false friends, untrustworthy leaders and rampant injustice is getting to be very much like our own.
So who is the Prisoner of Azkaban? Sirius? Yes, obviously. Harry? of course; he's been locked up for his own protection. The whole wizardly establishment, reduced to fighting evil with evil? Yes, indeed. The smarter question might be, who isn't?