In this Numberphile video (just after the three minute mark) it is stated that a collection of manifolds are 'not just homeomorphic, but also isomorphic'. It goes on to say that 'the changes required to get from one to the next do not require parts of the manifold to be moved through each other'. I have studied some differential geometry, but I have never come across the term 'isomorphism' used in this context.
My question is: what is the formal definition of isomorphism in this context?
My best guess is that this has something to do with the way these manifolds are embedded in $\mathbb{R}^3$, and is perhaps related to the concept of isotopy, although I'm not particularly familiar with this topic. For completeness, the manifolds in question are all 'deformations' of the 3-holed torus.
Remark: I cannot imagine that 'isomorphic' is being used here to mean 'diffeomorphic' --- it is known that all two-dimensional smooth manifolds which are homeomorphic are diffeomorphic.
I'm pretty sure he means 'isotopic', based on his description. Homeomorphisms are isomorphisms in the category of topological spaces; he seems to be a bit too sloppy on the terminology.
He also implies that 'isomorphisms' aren't homeomorphisms, but they are a special type. Not every homeomorphism is realizable via an isotopy in the manifolds ambient space. But it is realizable via an isotopy in some space (usually of larger dimension).