Does it have a screw that tightens on the shank?

No screw on the door knob

All knobs I've seen stay on by a set screw, look for a tiny hole in the stem of the knob. Hardware stores (not big box) carry replacements. Use an allen wrench to put the screw in aligning it with a hole or indention in the shank.

The bent end goes through the lockset, to prevent it from pulling free. There must be a setscrew on the knob to secure it and clamp the two halves of the shaft together. Perhaps someone replaced the knob with an inappropriate type. Lots and lots of vintage knobs on ebay, but you need one threaded for setscrew.

Casey

Upon further inspection and with your guidance I have indeed found a set screw. Here are photos of the knob and the exterior of the mortise lock where it is attached.

If that's pulling out, you need to heat the bent end white hot with a MAPP gas torch and bend it a bit more. The ones I have seen have a bit more hook to them.

Thanks. Is there also a way to tighten it inside the mortise lock?

Here's what it should look like if you don't want it to pull out on you:

Thanks very much for the link to the split spindle. I see the bend and conclude that the existing spindle may be worn out. If I were to replace all existing hardware what will I be looking for?

Picture of side plate, functions like a Baldwin Estate mortise lock.

Follow the link I posted in my last reply. It goes to a vendor of a new split spindle. The link is green, it says "new split spindles"

Thanks, I did follow the link, will order one.