I've seen this example given numerous times, but have never seen a real proof in a textbook.
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I am assuming that the questioner knows that the question is about either the surfaces or the 3-manifolds in question. A rigorous, yet diagrammatic proof is in our book Knotted Surfaces and Their Diagrams.
They are both homeomorphic to a sphere with one handle attached. This is quite clear for the coffee mug (where the handle is precisely the handle.....) and it is easily obtained for the donut (a.k.a. torus) by playing a bit with its representation as a square with opposite edges identified.