...is this Willy Daro or not?  

I couldn't really tell what was going on with the lamp until I turned the photo and lightened it a lot. You photographed in front of a mirror, right? So the 2nd socket at the top is just a reflection, and the large dark square is a painting or something.

OK, that aside---i don't know who designed or made it but I would assume the cord feeds into the hole in the base and up the hollow stem to the socket. If the bottom of the base is not open, then you will have to figure out how to get the end of the cord through it to the opening of the stem, or vice versa (feed wire down the top of the stem out the bottom). Usually there's a way to do it without driving yourself crazy.

Yes, crappy pics...

You are right with the second socket and the square.

The bottom is just open, so there is no problem to install a wire. But it looks crazy to me that there is no lid and you can see just the bare cables if you turn it upside-down.

Especially I think of Willy Daro if I look at the last picture with the socket. I did compare this area with other lamps and it looks very similar to me.

Metal lamp bases are pretty much always hollow, sometimes with a weight inside. All lamps are just going to have bare cables on the bottom without the felt -- the only thing you're missing is that piece of felt (sometimes reinforced with cardboard) that would be glued to the bottom. Beautiful lamp. Sorry no help on the designer/manufacturer.