Not exactly.
The center label stands proud of the playing surface. Drop one record (level) on to another and the labels contact first. They're made that way on purpose, mostly so if you set the record down on the table, the sound grooves don't get damaged (or at least give it a chance not to damage the grooves). There's also the air cushion that is formed between the two parallel surfaces that slows contact as the record falls. Won't do anything to accelerate the record, but it does lessen the impact force. Lastly, the sound is down in the groves, not at the top of the crests. So even if the vinyl contacts vinyl, it's highly unlikely to damage it.
Stackers are useful if you have a bunch of people over and want to have some music in the background. Stack up the albums you want to hear and let it go. Or, load up something like the Springsteen box set and let it play them all. When it's done, flip the stack over and play the other sides. Great if you're doing your chores around the house.
Certainly not an ideal setup though as the "stacked" record can get induced vibrations since it's not sitting on the platter mat but rather on a slippery surface. There's also the change in VTA as the stacked record requires the tone arm to ride at a different angle, changing the angle the needle rides in the grooves. This can change the sound reproduction in a couple ways, none of them good.
But damage? Far more likely to damage your vinyl with a worn or poorly aligned cartridge/stylus, not setting the tracking weight properly, messing up things like antiskate, or not properly cleaning them.
Me? I'll probably only use it once or twice. Just to prove to myself that it works. I'm like that with most of my vintage equipment (lathe, sewing machine, vehicles, record players, etc); if it had an option available when it was first sold or specific owner/service manuals, I want it in my collection, mostly because I just want it complete.