Why doesn't a conductor between two wires affect the current wires? When an aluminum plate is placed between two wires (lying parallel to each other), it doesn't change the repelling nor attracting force between the wires when a current is passing through them.
But when you encouter with a magnet a coil of wire there is a current induced in this coil and by Lenz'law the coil will produce an equal magnetic field which would repel the magnet. 
Now I should expect that this also would happen with the two wires, that in the aluminium plate an equal magnetic field is induced no matter what direction the current is. But this doesn't happen. Why not? Or did I mix things up?
 A: 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.)
A: 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
