The chuck only needs to be tight enough to hold the tap shank true whilst the square gets a grip. The square gripper is floating, not self centreing as such but so long as the shank is true it will come on centre.
That looks to be a solid body, direct drive, version. I personally would arrange it with some sort of fast drive release device rather than having it rigid in the machine. That way if the tap does start to bind you can release the drive and let things spin whilst you shut down in orderly fashion to clear the jam. Jammed tap + direct drive = snap. A carrier with a short sliding movement and pin drive so it releases when you draw back ought to do. As would a simple spring release friction clutch but getting sufficient drive for larger taps may be a problem. Depending on the machine direct drive tends to be rather fast too. Best to use proper machine taps too.
I'm spoilt in having a set of Pollard tapping heads (all but the very smallest) about the place. Internal reduction gear with automatic forward and reverse clutches so the moment you release pressure in either direction drive is lost. Mr Pollards tap holder is similar to the one shown but the chuck part is a simple three sprung jaws and screw down cone affair of the sort normally found on an ordinary turn the handle "eggbeater" drill.
Clive