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The answer depends on the context. Topologically, it's one. I personally like zero. If I say "There's a hole in my straw!" You'll not think all straws have holes. You'll think there's something wrong with it.
A cup is essentially a self-contained hole that we pour stuff in, but if I say there's a hole in my cup you'd know what I meant
From a topology perspective, a cup does not have a hole. A mug does, but its the hole that the handle makes, not the area containing liquid.
I see a lot of folks referencing topology, but clearly topology is not a particularly good go-to for how to talk about holes.
From a topological perspective, a hole in the ground isn't a hole... But you can still fall into it.
From a topological perspective, a hole in your logic isn't a hole -- but you can still have one.
Clearly we're talking linguistically, not topologically.