Here are some comments directed the bountied version with the prompt
Could someone please explain where the idea of stretching and bending relates to the ideas in topology? How did people start thinking of this intuitive in this first place.
This is clearly giving a more historical bend to the question, which seems to have had more of a philosophical flavor. The new formulation of the question is still too philosophical it seems to me (for better or worse); and there are already many nice answers addressing these, so I'll focus on the historical context of two (plus one) matters:
- "stretching and bending"
- That the surfaces obtained from two rectangles by gluing two opposing sides of each rectangle with $n$ and $m$ half-twists are homeomorphic iff $n$ and $m$ are both odd simultaneously (in which case they are both homeomorphic to the Mobius band), or even simultaneously (in which case they are both homeomorphic to the cylinder). This is the example that is referred to in the OP, and it sets up the issue: we have two surfaces both homeomorphic to the Mobius band, but it seems one can't stretch and bend one to get the other one.
Instead of building up to definitive answers to the questions above I'll only point to some sources.
First off, "homeomorphism" seems to have been used first in crystallography; its use in mathematics seems to be unanimously attributed to Poincaré. Here is the OED entry:
Indeed, to triangulate further see https://mathshistory.st-andrews.ac.uk/Miller/mathword/h/ , or more properly Moore's paper The evolution of the concept of homeomorphism" (http://dx.doi.org/10.1016/j.hm.2006.07.006). Of course Dieudonné's A History Of Algebraic And Differential Topology, 1900-1960 is also very relevant to the matter at hand.
Thus the first use and definition of a "homeomorphism" is to be located in the Analysis Situs (p.22-23 of https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf) (by which in fact a "$C^1$ diffeomorphism" is meant).
According to Moore's paper, Poincaré himself changed what he meant by a homeomorphism over time, and without acknowledging that he was doing so.
Again according to Moore's paper, the idea that looking of invariants of "stretching and bending" was already familiar to mathematicians before Poincaré, e.g. Möbius (for this he cites Pont's La Topologie algébrique des origines à Poincaré; Dieudonné also points to this work as the definitive source for topology-before-Poincaré)
Despite the fact that Poincaré was the one who defined a homeomorphism (as a diffeomorphism), item 3. in my list above should have been addressed by the time of Poincaré. Indeed, according to the digital copy of a translation of Analysis Situs (p.2), von Dyck had already classified non-orientable surfaces in 1988, despite the fact that in fact von Dyck was using a continuous bijection as an equivalence, according to Moore's paper (the orientable ones were classified earlier Möbius). I am extrapolating that the "parity of the half-twists" issue came up (von Dyck's paper is available online to verify or debunk this).
Finally let me mention the "parity of the half-twists" issue has been discussed multiple times here on M.SE, see e.g. (some of these are marginally related)
On a personal note, I remember being puzzled by this matter when I first started learning topology. I also remember looking at some of the related M.SE discussions above. Still, it didn't make sense to me that the way a circle is the same as a square was to be considered the same to the way a Mobius strip with one half-twist is the same as a Mobius strip with three half-twists. As some of the discussions show above that indeed, eventually there is a difference (existence/nonexistence of an ambient isotopy) which is topological (i.e. it's a difference that can be articulated using only the language of topological spaces and continuous maps), but this difference is not as simple as the existence or nonexistence of a homeomorphism between.
It seems the point inherent to all this is this:
- Different mathematicians (or same mathematician at different times) use different phrases and nomenclature to signify certain intuitive ideas (mistakenly at times). Often these intuitive ideas don't exactly work in tandem with each other, nor with the formal definitions. Despite this, it's still good to be aware, and knowledgeable about the sedimentation, if not to appreciate it.