Consider Mobius strips (eg. strips of paper with one or more half-twists joined together at the ends--for example, a Mobius strip with one half-twist has the interesting property of having only one side...). As anyone who has had a chance (either as a child or as an adult) to play (i.e. to cut them down the middle lengthwise for example) with such objects can verify, these objects have very interesting properties. It also seems a trivial observation that these strips of paper (or any other material that can be so twisted) so constructed constitute real mathematical objects and therefore mathematical Platonism is true. What, if anything, is wrong with this argument? Another question I find of interest regards the half-twists that give these objects their interesting properties. What sort of property are these half-twists and what are they properties of since any material that can be given half-twists and can be joined at the ends become these 'real' mathematical objects by the operations of twisting by half-twists and joining at the ends?
Mobius strips aren't that interesting physically because they're all just rings. Everything is three-dimensional. If you took a noodle and squashed it so it was 30% wider than it was tall, and instead of doing it flat you did a half-revolution so that one edge of the bulge went from left to up to right, you'd have a "Mobius strip" of a sort. But it's a really unimpressive one because the curvature is still nearly constant as you travel around (rather than along) the noodle.
Doing it with paper just makes the aspect ratio more and more extreme. You can squash your noodle into a Y shape had have it rotate by 120 degrees also to get an interesting shape (if you forbid yourself from going over the high-curvature "edge"). I don't think any of this proves anything about the reality of objects, but it demonstrates that practical geometry is fun.
Some of your questions might be better addressed on Mathematics.SE. However, I would address the philosophical content of your question as follows.
It also seems a trivial observation that these strips of paper [...] constitute real mathematical objects and therefore mathematical Platonism is true. What, if anything, is wrong with this argument?
Just because a physical object can be described well by a model, consisting of a mathematical "object" which one might describe as having "very interesting properties", does not imply that the mathematical object is any more than an abstraction with which you can describe the physical object (or a class of similar physical objects).
This is especially clear when those very interesting properties are made with reference to imaginary physical models. What does it mean to cut the ideal Möbius strip down the middle, when you don't have manual access to it and it is in any case presumed to be eternal and unchanging? What you're doing is imagining the Platonic ideal as if it were a physical object, or at the very least imagining a model where certain details are fixed — the Platonic Möbius Strip That Is Made Of A Continuous Strip Of Some Quite Flat And Easily Cut Material — in order to express properties about physical models which you would somehow like to ascribe to the Platonic ideal.
The problem with the argument is that you equivocated between the abstraction and the real; between the map — represented by The Möbius Strip, with capital letters — and the territory, which is a single instance of a strip of paper cut and twisted so that the most natural description of it in terms of metric spaces (as opposed to material science) is as a finite non-orientable 2-manifold of a certain well-known kind which we call "Möbius strips".
No, not particularly. What it shows is that the idea of two sides is more subtle than one may believe without reflecting on an example as you've pointed out.
For the instrumental definition you've given it is true that the strip has one side.
But why is it confusing to someone that hasn't come across a mobius strip? This is because there's alternate definition of sides. To wit, go down through the material of the mobius strip and if you emerge again then it has another side. If you had an infinite thick strip for example, you wouldn't be able to get to the other side.
All this shows is the ambiguity of definition.