Bob wins.
I'm assuming the eligible pieces are the knight, the bishop, the rook, and the king.
If Alice picks the knight, the rook, or the king, Bob can win by picking the same piece. On each of his turns, Bob can play the move
that is the 180 degree rotation of the move Alice plays. This works
because the starting position of the board is rotationally symmetric,
so after Alice's first move, Bob can make the 180 degree rotated move.
This puts the board back into a symmetric position (including
forbidden squares), allowing Bob to mimic Alice's next move in the same way. Eventually, Alice will run out of moves.
This strategy will not work if Alice picks the bishop. If Bob also picks the bishop, then on Alice's first move she can move her bishop
across the diagonal and place it next to Bob's bishop, preventing Bob
from moving at all on his turn.
If Alice picks the bishop, Bob can win by picking the king. Assume that the players put their pieces on the black diagonal. Bob's strategy is to advance the king along the black diagonal toward the opposite
corner until this is no longer possible. At this point, Alice's bishop becomes trapped one one side of the board, while Bob's king can take advantage of all the black squares on the other side, plus all of the white squares, which Alice cannot move to. Alice will run out of moves before Bob does.
A note about the rook and bishop cases: Alice is able to thwart the symmetry strategy in the bishop case because she can move multiple squares directly toward Bob, moving through the square Bob needs to occupy. For example, say Alice's bishop starts on a1, and Bob's on h8. If Alice moves to g7, Bob's symmetric move is to b2. However, Alice moved through b2 to get to g7, so Bob can't move there. This cannot happen in the rook case. In order for Alice to move her rook directly toward Bob's, the two rooks must occupy the same rank or same file on Alice's turn. However, Bob's strategy ensures that the board is always rotationally symmetric at the beginning of Alice's turn, and no position with the rooks in the same rank or same file is rotationally symmetric.
The symmetric king case does not fall into the same trap as the bishop case, either. Although Alice can move directly toward Bob along the diagonal, she can only move one square at a time. In any rotationally symmetric position with the two kings on the same diagonal, there are an even number of squares between them, so if Alice moves closer to Bob's king, Bob will be able to make the corresponding move toward Alice's king.
