Why doesn't a conductor between two wires affect the current wires?

the plate has 3 dimentions, height, width and thickness. And In the setup of the experiment i've seen ("for the love of physics" on youtube)

• height >>width>>thickness
• the plate height is being parallel to the length of the wire
• the plate edges separated by the plate width are equidistant to each wire

As a result the induced current loops in the plate during the wires turn on/off occur mostly in the thickness of the plate. These eddy current loops being contained in size by the small thickness have little effect on the magnetic field

Imagine the set-up of your two wires and a sheet of metal in between them. The electrons in the metal are static (for our purposes), hence the Lorentz force they experience is $boldsymbol v times boldsymbol B = 0$. So the electrons stay static, and static charges don't generate a magnetic field. In other words: while it is true that a metal shields electric fields, it doesn't shield magnetic fields.

(As a comment to your note about a magnetic field being induced in a coil near a magnet: only if you move the coil; if you have a static coil near a magnet, the coil has no induced magnetic field.)