Am I to understand this is not a regular screw?

My apologies, the second pic did not register in my head until after I posted that. It appears that register came with it's built in latches and a quick internet search reveals you are not the only person to run into this problem. One guy just rebuilt his. Unfortunately I will not be of any help to you. Just curious though, how old is the house?

The latch is probably one that turns back and forth held on by rivet type connection that allows user to slide back and forth.

If you're skilled with tools, you could take a drill with a metal drill bit slightly larger than the rivet holding that clasp to the register.

Use a wide tooth flat screw driver to pry that clasp from the vent register --- the back of the vent cover, then drill that rivet from back side of register to remove that sliding clasp, then drill a somewhat smaller hole than the screw size you plan to use into where the clasp slides in on the vent side in the wall (floor or ceiling --- however the vent boot is mounted)

Choose a screw (lag bolt or other of your preference) and use that to keep the vent register in place.

Otherwise, you're looking at sizing up a new register to fit over the existing boot of the vent in question, not to mention figuring out a way to attach it.

If the vent cover and frame are steel, use some strong rare earth magnets to hold in place. Depending on how thick the magnets are and how much space is between the frame and vent cover, you might need to glue some thin spacing material to the back side of the vent cover and glue the magnet to the shim.

Or use some wing clips (single wing like are used to hold old wooden frame window screens, etc. Remove the existing twisty thing on the vent cover, insert a bolt through the hole, thread a nut and then a wing clip on the back side, adjust it to the proper position to hold the vent cover tight to the ceiling (trial and error) and then epoxy the nut and wing clip to the bolt to make your own (crude) twist lock.

If I read the picture correctly they show the fastener on both sides of the cover but don't show what it connects to on the duct. It looks like the fastener had a tab that went into a slot and a 1/2 turn of the fastener would either engage or disengage the slot. You could cut the fastener off with a dremel tool and then jury-rig a different method of making the connection using a regular screw.

Just drill 2 or 4 holes in it and fasten it with sheet metal screws. Or clean thoroughly and use heavy duty stick-on Velcro.