A pizza is not really a pie although it's sometimes referred to as a Pizza Pie.
Pizzas are not really a pie in the same way as a fruit pie or meat pie is although they are sometimes called a pizza pie.
The term pizza pie is more of a descriptive phrase that refers to the round shape of the Pizza and the crust and not a classification as being a pie.
Even in Italian a Pizza is mainly seen as simply being a Pizza and not a pie.
When it comes to what qualifies as a pie, the pies are defined by their crusts.
For example a filled pie (also single-crust or bottom-crust), has pastry lining the baking dish, and the filling is placed on top of the pastry but left open.
A top-crust pie also has the filling in the bottom of the dish and is covered with a pastry or other covering before baking.