OP, there's a pretty good variety of monitor mounts (and monitor stand mounts) available on sites like Amazon. For example,
here's one to illustrate the concept. I chose to "hang" the heavier Mac Studio on a wall behind my monitor in something like those and it has worked just fine for over a year. Bracket to wall and some velcro straps through the bracket wrapping around Studio in a cross pattern so it can't possibly fall.
Post #2 rec will cost less and you can zip tie all the way around the Mini, making something like a zip tie "cage" if you will. In other words, use velcro to attach it to the stand and then maybe some zip tie all the way around for some added confidence the velcro (alone) won't fail you. Get "heavy load" velcro vs. best price velcro.
However, for $10-$30, I'd just get a dedicated Mac Mini mount to avoid the glue of velcro and add the confidence of a solid structure that could later be removed as if it was never there.
Two other options to consider:
1. How about a riser with a slot big enough for mini like this one from Twelve South...
There's many of these shelf (and no shelf) monitor risers available and Mac Mini will tuck into/under many of them.
2. How about a vertical stand/hub, which delivers even easier access and more ports...
There are several of these available for Mac Mini too. And whether stand or riser, it can be pretty nice having much easier access to ports than having Mini stuck to the back of a monitor.