We are all making the assumption that the register has zero run out compared to the spindle bearing diameters.
If the spindle nose is showing zero run out, the 0.05 mm is within whatever you fit to the spindle.
To have a 3 jaw chuck that holds work concentric within 0.05 mm (2 thou) suggests that it is a good one.
To engage any chuck on the cylindrical register, there HAS to be a finite clearance, probably about 0.05 mm. You would have a struggle fitting or removing if register on spindle and chuck were size and size!
If you think about the clearance necessary between register and backplate and then the clearances within a 3 jaw chuck (individual jaw / scroll, scroll/body) getting work held to less than 0.127 mm (5 thou) run out is pretty good.
If you want to have work absolutely concentric, then use a four independent chuck and keep clocking and adjusting until the eccentricity falls within your standards of acceptability.
If you really want to work to tighter tolerances, you had better have your workshop temperature and humidity controlled, like an industrial Standards or Calibration room.