The hole through the center makes a hole around it inside the mug.

Yeah, the hole through the center is one hole (the "donut hole"). The handle is the other (the "handle hole").

Topologically, the lip of the mug is a boundary, not a hole. This object is of genus 2 given it's a 2-manifold.

Edit: The explanation below is the product of a brainfart.

~~But we can show this assuming the lip isn't an abrupt boundary and that instead that the outer wall of the inner loop is part of the same side as the outside of the mug (i.e. that you can continuously walk on the surface from the outside past the lip without crossing an abrupt boundary, which more closely matches the physical reality of this mug).~~~

Watch this: imagine a loop around the hole of a donut. Now try to contract that loop down. Without cutting it, you can't.

Now take a loop around the lip of this coffee mug. You can contract it to a point by bringing it all over the lip of the mug, then inside, along the outer walls, and then finally down to the bottom of the mug. It all meets up there. No hole. There's the donut hole and the handle hole, but the colloquial "hole" is either a boundary if the lip is flat or not even a boundary if it continuously curves inward.

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.

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

but

It's solid inside rather than hollow.

Shit, you're right. I was thinking of a normal mug. So "adding the donut hole" actually created two holes.