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But here's the thing. Take that doughnut and stretch it until it's a cube with two square cutouts in it. Stretch in some of the inner walls. Now you have a house, with a door and a window. Now: does the house have two holes - a door and a window - or does it have one hole?
Locally has two extrinsic holes, that is holes relative to things outside and inside the house, globally has one intrinsic hole. We say that the door is a hole respect to the wall no to the house itself. So both the door and the window are holes locally. But we never say the house has holes, we talk about walls and ceilings so globally that house has 1 hole. Another way of thinking it is that if the house can be deformed into a filled doughnut then it can be compressed to a circle and that's the definition of a 1-hole.
Topologically, still one