Topologically, that part is not a hole. The liquid basin not being a hole is where the joke in the OP originates, even: a standard coffee mug with a basin for coffee and a handle is homeomorphic (topologically equivalent) to a torus (a donut). And that’s because the handle is the hole, not the basin.

In the case of the OP, ignore the handle for a second just to simplify things. That’s a hollowed-out donut (torus) except that you’ve taken a part of that donut’s surface and cut it out. That area that’s been cut out isn’t topologically a hole; instead, you’ve just created a boundary on the 2-manifold (read: flexible surface).

Topology has rigorous, algebraic definitions under the hood, but that’s what’s going on in this picture topologically: you’ve taken a donut, glued it to another donut, and cut a “hole” (colloquial usage) in the double-donut’s surface, creating a boundary on the surface.

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Edit: To hopefully justify this a bit intuitively for the standard coffee mug (not this abomination in the OP), the utilities problem is a classic toy problem in topology that

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is unsolvable on a 2D plane

but

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can be solved when you embed it into the surface of a coffee mug.