Agreed that it doesn't really look like a Morse taper. Also agree that it looks a bit rough (grubby anyway). And as someone else already said, all those I've seen are "shiny" (ground) all the way down.
If CHJs tips don't work then I'd A) talk to Axi, and/or B) look for the "missing" machine screw (up inside the chuck with the jaws open wide) which MAYBE missing (most do not have or need them). In any case, such machine screw, IF used, is normally used to hold the chuck body onto the adaptor which goes up into the tapered hole in the pillar drill (quill). The whole (sorry!) idea of tapers like Morse, Jacobs, etc,) are that provided there is NIL damage on the two faces, then the whole (sorry again!) thing is firmly held together simply by the interference fit between the adaptor (male) and female (quill).
Big bash/es to retain the chuck and its adaptor into the quill should NOT be necessary. Just a single firm "clunk" with a big piece of wood up from under should do it, and as soon as you come to drill the 1st hole after refitting, then the whole lot should be a complete, very firm single assembly.
For the same reason, personally I disagree with the idea of using Loctite (or similar) bearing retainer "glue". If the 2 parts go together OK as CHJ has described, and assuming they're both perfectly clean, then bearing retainer will certainly hold everything together OK, but that shouldn't be necessary. And if using a bearing retainer you'll have quite a big job to ever get the assembly out of the quill again - even if your pillar drill does have the typical slot in the quill with tapered drift to gently knock the assembly out again.
