I'm not a physicist, but I am in the STEM field. I think that the induced magnetism in the rod is due to the proximity of the other fields. It doesn't have to be moving or touching. Surround a strong magnet with non-magnetic materials and its field extends a ways, so you can then stick a rod next to it (not touching) and that will focus the magnetic field and make it locally stronger than it was without the rod.
I don't know how else to explain it, but it's a thing. Sorry if I wasn't technical enough. I'm on my phone.
Edit, oh yeah, so if you move the metal relative to the magnet, you will induce a current due to the change in flux.
In order to maintain the magnetism after you remove it, you'd have to run it through this field many times. Rubbing is just so you get maximum strength but you don't have to physically touch the other magnet, so that the "iron atoms align" which is all I could find online.