The genus is insufficient to determine if an object is equivalent to the other. Orientability distinguishes the mobius strip and the torus, a torus is orientable whereas a mobius strip is not. Therefore, topologically speaking they are not equivalent.
you can make a mobius strip with paper. then get a pencil and try to orient it in the mobius strip. that is, make it normal to the paper then move it around. you will see that if you go though the strip and go back to the starting point the pencil will be in the other direction. thus, the orientation is not continuous so the surface is not orientable