NOT a pie.
Ah yes, the question of "why?" The eternal six year old child in all of us is never satisfied being told that a thing is a thing. We always need to know why. Step forward into my private mind chambers, in my memory palace. Mind where you step though, I've had the trapdoor to the pit of insanity re-oiled.
If we're going to say that a thing is not a thing, then this is very much an ontological question on the nature of being. Before we go there, we need to be aware that names are deceptive. A Boston Cream Pie for example, is very obviously not a pie because in my not very well paid opinion, experience tells me that they are all cakes. If this be true that a name is no guarantee that a thing is or isn't a thing, then a cheesecake could very well be a pie.
What is a pie?
The short answer is that a pie is a bowl shaped layer of bready material (and I have to include all possible pastries, crumbled up cookie things, and even pasta) into which some other stuff is poured. That bowl shaped receptacle of bready material is generally called a crust or a shell.
That working definition is broad enough to include all manner of sweet pies, savoury pies, and historical abominations which are insane combinations of both.
Using this as a starting point, the things that we generally accept as pies are obviously pies. Fruit pies including Apple Pie, Cherry Pie, Apricot Pie, as well as Steak & Kidney Pie, Chicken Pot Pie, and variations on a Meat Pie which include curry, chili, bacon, cheese, carrot, tomato, a whole host of gravies and sauces, are all included.
I don't think that the top of the pie needs to be defined at all because a Shepherd's Pie which has a layer of potato on the top, as well as fruit pies with a criss-cross lattice of pastry, are already both obviously pies.
This does of course immediately beg the question of whether or not a Quiche is a pie and again, the answer is obviously "yes" because by definition, a Quiche is a bowl of pastry into which what amounts to an uncooked omelette is poured and then cooked. Quiche is a pie.
Also, this begs the corollary question of whether or not things put into a ramekin with a lid of pastry on top is a pie and I would argue that they are not. They fall into another category with the amusing portmanteau name of a "stewpé" because they are basically a stew with a toupé of pastry on the top.
Naturally this leads to the delicious abomination of the Scottish Restaurant's (we dare not speak it's name because that's bad luck) Hot Apple Pie. Is that thing a pie? Amazingly, yes. You have a rectangular bowl of pastry into which stuff is poured, then covered over with another layer pastry (albeit identical) and then even though the whole thing is deep fried it's still a pie because the method of cooking isn't an integral component of the ontological properties of being a pie. That also goes for its siblings the Hot Cherry Pie, Hot Custard Pie and Hot Apricot Pie.
Back to the opening question. A cheesecake is almost always a layer of crumbled up cookie things on the bottom onto which a layer of sweet cream cheese mixture is put, and a myriad of variations follow. I have never ever even once seen a cheesecake where the layer on the bottom came up the sides to form a bowl. Admittedly one could build a cheesecake inside a pastry crust but immediately you'd want to call that thing a Cheesecake Pie.
I will also admit that cheesecake could be put into a bowl like thing as per a tart but then you would have a Cheesecake Tart, which is probably as close as you get to a pie without actually being one.
Cheesecakes are of the same kind of form as Chocolate Bavarians, which are also not pies. If we start including things which are round, have a base and can be cut into slices, then this is the way of madness. Start on that path and Pavlova will want to barge its way onto the stage of pies and then there's the question of whether or not Custard or Caramel Slices are pies as well; which they clearly are not.
Cheesecakes are not pies; just because they have a base of crumbled up cookie things doesn't help their cause either. The thing that prompted this and why the question even needed to be asked was because a client of mine saw Cheesecake in the section of a menu at a coffee shop called "Pies". It's a good thing that I don't know which coffee shop it is because as a long term resident in the house on Pedant Corner, I want to mark on that menu in big red letters "Cheesecakes are not pies. See me after class."
I'm not casting ill faith on cheesecakes because they are yummy and that isn't in doubt and I don't want to be pastry racist because all of them are delicious but in the world cup of food nations, Cheesecake isn't eligible to play on Team Pie because it isn't a citizen.