The elevator vendor will choose a 3-phase induction motor.
I'm not sure why you're worried about the motor selection, but it will be what it will be, the only influence you have is in the specification of the capacity and speed. The elevator vendors have design consultants that will work with you to select an elevator and they will then provide a package that specifies what the building needs to provide in terms of space, structure, power, and HVAC. If it is a hydraulic elevator the motor will near the base of the elevator. If it is a traction elevator the motor can be at either the top or bottom of the shaft, but motor at the top is far less expensive than motor at the bottom; that is also something that you can specify, but probably doesn't change the motor that the elevator vendor will select and supply.