I think you should go to the kitchen and count them

A mug has one hole, as do a donut or staw. Topologically speaking, the “hole” you put your liquids in is not a hole, as it does not go all the way through to the other side. The handle is the only hole.

Demonstration gif: https://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif