Verifying 'standup maths' homeomorphism claim

They are not homeomorphic. They are merely homotopy equivalent.

A way to see they are not homeomorphic is that they have different numbers of boundary components (three versus one).

A fancier way (using homology) is to consider the fact that a sphere with three holes can be embedded in the plane, which implies that the algebraic intersection number of any pair of closed loops is $0$ modulo $2$. But on a torus with one hole, it's easy to come up with a pair of curves that intersect in exactly one point, which means the algebraic intersection number is $1$ modulo $2$.

Addressing "they have thickness": if we are considering (as Parker does in the video) thickened surfaces, which you might formalize as products of a surface with an interval (and thus are 3-dimensional manifolds), then the thickened surfaces are indeed homeomorphic. They are both genus-2 handlebodies.

In general, if an orientable surface deformation retracts onto a wedge of $g$ circles (like $S^1\vee S^1$ for $g=2$) then their thickenings are homeomorphic to a genus-$g$ handlebody.

There is another way you can formalize a thickening that depends on the way a surface is embedded in $\mathbb{R}^3$, which is to thicken it into the ambient space (i.e., take a product with the normal bundle, rather than with a trivial $I$-bundle like above). You can drop the dependence on orientability with this notion of thickening. An "ambiently thickened" Mobius strip is homeomorphic to an "ambiently thickened" annulus, where both are homeomorphic to a genus-$1$ handlebody (a solid torus). But thickened in the first way, they are non-homeomorphic. The Mobius strip gives a non-orientable 3-manifold, but handlebodies are orientable.

Watching the video, he seems to be aware that they are not homeomorphic as 2-manifolds.

He says that to justify the crucial step of going from two perpendicularly glued cylinders to two annuli joined on the side within a plane, you have to consider the thickness of the fabric, so that they are 3D objects, rather than the thickness in going from 1D (2 joined circles) to 2D.

See this point in the video.

There is a loop in the punctured torus whose complement is connected but the complement of a loop in a sphere with three discs removed is never connected, so they cannot be homeomorphic.