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?
 A: If he has a friend then they both can paint their right hands blue and left hands red.
His friend stays where he is, he goes once around the strip, now his left hand and right hand are switched when he compares them to his friends hands.
A: 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.
A: Flatlanders of Relativity theory as we know can measure motion with successive motion derivatives to get information about spatial orientation on an inertial frame. 
Torus/Möbius Band homeomorphism 
A slender Möbius band is very different from a bulky Möbius band of rectangular or near square section, please note. You get  orientability  and navigation without external reference  when you dare to look and go sideways around its edge or cross the border under your feet through a hole in the band taking a short cut.  
EDIT1:
In the link below, I observed that as the Möbius Band gets fatter, it gains in orientability due to increased access though or across its thickness. 
Torus/Möbius Band homeomorphism
One take away view from there is that the Möbius band is very different from the fattened Möbius band; the non-orientability of the former doesn’t imply anything about the latter.
If inertial navigation is possible by local measurements made by FlatLanders, imho, it removes sense from the notion of orientability of the Möbius Band.
A: Not about the Möbius strip specifically, but about the orientablility of the space: in The Wall of Darkness by Arthur C. Clarke the inhabitants of a planet discover that their universe is nonorientable. See What is the geometry of the universe in “The Wall of Darkness” by Arthur C Clarke?
