Haroldg48 is absolutely correct in that being marked "Solingen" is not itself a guarantee of quality. And looking at the first pic it would appear to be a lower-quality razor. For example see the tang near the pivot. On a well-made razor there would not be any marks like can be seen there. At least not without heavy (mis)use.

But beyond that, even if a razor was made from the best steel in the world by a master craftsman, it will not cut properly unless it is honed properly. That would be my first guess: it isn't cutting because it simply is not sharp enough. Does the vendor specialize in straight razors? Or is it a more general shop (knife shop for example) where staff may not appreciate that true razor sharpness is a whole other level from knife sharpness?

It sounds like it has been unevenly honed as well. The edge should cut equally well along the entire length of the bevel. But it sounds from your description that this is not the case. So suggestion #1 would be to have it properly honed by a specialist in razor honing.