Why is stress a tensor and pressure not?

Stress is a tensor because it describes things happening in two directions simultaneously. You can have an x-directed force pushing along an interface of constant y; this would be σxy. If we assemble all such combinations σij, the collection of them is the stress tensor.

Pressure is part of the stress tensor. The diagonal elements form the pressure. For example, σxx measures how much x-force pushes in the x-direction. Think of your hand pressing against the wall, i.e. applying pressure.

Given that pressure is one type of stress, we should have a name for the other type (the off-diagonal elements of the tensor), and we do: shear. Both pressure and shear can be internal or external -- actually, I'm not sure I can think of a real distinction between internal and external.

A gas in a box has a pressure (and in fact σxx=σyy=σzz, as is often the case), and I suppose this could be called "internal." But you could squeeze the box, applying more pressure from an external source.

Perhaps when people say "pressure is internal" they mean the following. σ has some nice properties, including being symmetric and diagonalizable. Diagonalizability means we can transform our coordinates such that all shear vanishes, at least at a point. But we cannot get rid of all pressure by coordinate transformations. In fact, the trace σxx+σyy+σzz is invariant under such transformations, and so we often define the scalar p as 1/3 this sum, even when the three components are different.1Now the word "tensor" has a very precise meaning in line