I'm out of my depth but I believe it's a torus because it's 2 cylinders, not circles?

A cylinder is a circle and a line. The circle goes around and the line is the height.

A torus is a circle and a circle. One circle goes from the outside to the inside and back around, and the other goes around the outside, if we're talking about a donut. We can see that any point on the surface of the donut is some combination of "go around the outside" and "walk around towards the center". You can also imagine taking a cylinder and bending it around to make a torus -- this changes the 'height' line into a second circle.

We're talking about the product space, so what that means is you can describe the space by (two) coordinates, one drawn from each "shape" (actually, manifold). A "torus" then differs from a "plane" in that you can wrap around in the x-direction and wrap around in the y-direction, independently, since the x-direction on a torus is a circle instead of a line and the y-direction on a torus is a circle instead of a line. By contrast, a cylinder you can only wrap around one way, because you can only wrap in the direction that has a circle for its coordinate space.

You cannot take a piece of paper (of any size) and wrap it around part of a donut.

I'm not sure what you're trying to say, could you elaborate?