Side play is normal. As long as the wheels don't rub it's O.K. The side clearance is from clearance in the turbo center housing and the bearing. Typically the shaft to bearing clearance is .0015" to .002" The bearing to housing is about .0015" to .003". When you add the clearances up you can see you will have some slack. The engine oil pressure will "float" the bearing and shaft. The assembly will then center itself and spin true to center.
I use to build turbos. We had one application on a rather larger turbo that always felt sloppy due to factory tolerances. I finally cured the rebuilt sloppy problem by Installing the bearing with Vaseline. Never had any more loose turbo shaft complaints. Vaseline took up the slack till they fired'r up.