Is it common to refer to the Möbius strip as a 2D surface? You can't uniquely identify each point on its surface using only two dimensions, so I wouldn't call it a 2D surface at all. It lives in 3D, therefore it is a 3D surface, no?
Yes you can. Run a sharpie along the bottom edge of the paper so that it bleeds onto both sides of the paper. Now tape the ends of the strip together with a half twist to make a Möbius strip. Place a point anywhere on the strip. Draw a line through that point such that the line meets both edges of the strip at a right angle. Measure the distance along that line between your point and the darkened edge; call this x. Now, hold the Möbius strip in your left hand so that you're pinching it by the tape. With your right hand, run your finger along the strip, starting from the tape and moving to the right. Measure how far you have to move your finger in order to reach the line you drew; this could require up to two loops around the strip. Call this distance y. The tuple (x,y) uniquely identifies your point.
I say it's a 2D surface, because while it lives in 3D, it's "thin" in one dimension. It's like saying a piece of paper is a 2D object.
And when I say "it can't live in 2D", I'm kind of lying - it can live in a weird kinky 2D space, but NOT a boring 2D plane. Well, you could smash it into a 2D plane, but it would self-intersect (get all smushed together, losing its shape). And by "shape", I mean topological properties ... but I hate jargon.
But a Mobius strip is a 2D surface which can't live in 2D (a boring flat 2D plane); in the same way a Klien Flask is a 2D surface which can't live in 3D; and a knot is a 1D surface which can't live in 1 or 2D. Press a knot into 2D, and it crosses itself (self-intersect). Force a knot into 2D, and it will break, or join up with itself (depending on the material).
Real mathematicians (i.e. not me) will say "manifold", not surface, but it's more or less the same thing.