While cake specifically refers to baked doughs, the term has been extended to include any sort of rounded *thing* (e.g. urinal cake. Unsavory example, but first that came to mind). And a cheesecake is a round, flat thing, often served with dessert, so the description as a cake is sound.
Pie’s typically have a pastry crust whereas cheesecakes have a cookie or graham cracker crust.
Alton Brown covered part of this on Good Eats. Cheesecake is a custard pie, as its structure comes from egg and dairy proteins, and typically there is a crumb crust on bottom. As such, one can’t expect a cheesecake to behave like a cake during the baking process. He didn’t, however, cover the linguistics.
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