In Topology, you're allowed to move surfaces around much more freely than you can in the real world, so don't use your intuition from there, which says that coffee cups are too hard to move around. The point is that you can move the surface around in ways that in topology are considered to be no change to the surface, and get from one shape to another.

If you do want to use your physical intuition, a play-dough coffee cup can be squished into a donut without ever poking a new hole through the playdough and without ever closing one up. That's probably not great mathematical intuition since it still keeps some constraints that topology doesn't, but if you're not planning on studying topology, it's harmless enough.