which damaged part causes on a turbocharger to have shaft play?

 

bearing and thrust bearing

 

shaft play in which direction?

 

Side to side

 

Side to side is journal bearing. In/out is thrust bearing, but can also be from the thrust collar being worn too. There are other things that can cause thrust play like in turbos with bolt-down thrust bearings that come unbolted, or other oddball ---- that happens.