Curious... a really good piano player... can he play NOTE-FOR-NOTE identical when transposing to another key? As in, if the playing where put to staff music, his playing in key of Eb would be identical to key of E if they were both transposed to the same key?

Or does the geometry of the keyboard change things up a bit?

Say, you a session cat and the artist wants things dead on. Song is arranged in key of E but he can't sing it that day in that key, so, it's lowered to Eb. Piano player can nail those exact licks? Would he be able to do this in every single key?