Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip? Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity of a lower dimension is non-orientable? So it always takes a next dimension to prove that an entity of the current dimension is non-orientable?

If the person is in a Möbius strip, then it seems we are assuming he is $2$-dimensional.  Suppose he has with him two identical circles split into sectors of $120^{\circ}$, and each sector is colored a different color.  Notice being $2$-dimensional, he can rotate this circle but not reflect it, so the two circles are identical up to a rotation.
Now, let him leave one circle at a point, and wander around.  If he ever returns to the point where he left the first circle, and he finds that the two circles cannot be rotated to match each other, then he knows he is living on a non-orientable manifold.
