The reason why the paper Möbius strip is twice as long as the paper strip used, is, as far as I can underständ it, that paper has two more sides, apart from the two dimensional. A common paperslip cut for making a Möbius-model has long sides, lets call those "East-West" and short sides "North-South". But paper also have an underside or back and an upper- or frontside. Those two outfacing sides or surfaces are kept together by paper pulp. When the paper strip is twisted around its long axle, and the ends are glued together to a sling, it is not only the "west" sidewhich is glued to the "east", but also the underside of the paper to the upper side. Isn't what is really is made by the paper slip method a portrait of a 2d object in three dimensions, which differs from the original in several ways? And by the way, doesen't the edge of the paperslip Möbius also have a surface, which also consists of two Möbius-bands in a row?
Is'nt the paper Möbius strip a 3d object that is very different from the 2d Möbius band?
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When we make models of mathematical objects they are never accurate. They are supposed to represent the important features of the thing we are modeling. If we draw a circle with a compass, it is not perfectly round and the line has some width. I saw a fun discussion about how many dimensions a garden hose has. From a ways away it looks like a one dimensional object. You get closer and can perceive the diameter, making it a two dimensional cylinder. Closer still and you can see the thickness of the wall, making it three dimensional. By now one of the dimensions is infinite, not finite.
When we make a model of a Mobius band out of paper we are supposed to only look at the large surface. We are not even supposed to notice how it is embedded in $\Bbb R^3$ or that the band is two sides of a single piece of paper. It is a two dimensional finite region with certain connectivity.
You are correct that we could see a model Mobius band as a rectangular tube if we pay attention to the thickness of the paper. It then has two sides and two edges. For some applications that can be a useful way to look at it. I have read of conveyor belts that were given a half twist so they would wear equally on both sides (or the only broad side). Here you would care about the thickness.
Yes, considering the strip of (thick) paper as a bar with a rectangular cross-section, then it seems such a three-dimensional object, half-twisted and with its ends joined together, yields two Moebius strips, each one twice as long as the bar from which they are formed.