Defining them doesn't require any higher dimensions. If you have an image in the plane, you could say that your top is the image, and your bottom is what you get when you reverse all of the x-coordinates: a mirror image. It is silly though to call it a "top" or "bottom". You could scramble all the points around if you wanted. A mathematical surface doesn't require any notion of points sticking together like a plastic sheet, and you can permute the points just fine and still have "an image" in the plane.
If you want to talk about rigid motions, or motions where all the points stay "stuck together" with their "neighbors" then of course there are some things you can't do.
A mathematical object can indeed have a frame of reference. If I number the sides of a cube then it matters which one faces up. It doesn't matter whether I call it "1" or "top" or "purple". There isn't necessarily a preferred frame of reference- unless you define one, then there is.
A mobius strip doesn't need to be imbedded in a higher dimension. You can think about it as a type of rotation. Usually when we go around a circle, we arrive where we started after one rotation. On a mobius strip, after one rotation, everything is inside out, and after another rotation we get back to where we started. It's kind of like having to turn 720 degrees to face the direction you started in, instead of the regular 360.
spinors have this property, for example, even in three spatial dimensions. Don't bother trying to visualize it. My comment is that it would be helpful to study some fundamentals of analysis and topology and maybe some abstract algebra before you start trying to think about difficult objects within mathematics.
Nothing in mathematics "just is". We establish a set of first principles and deduce true statements or allowable changes from them. A problem with your question is that it doesn't really allow any sort of insightful answer because nothing in it is defined precisely.