I am using photoshop elements 15 on a system upgraded to Big Sur without problem. (Big Sur requires 64 bit) But when I went to install PS elements 15 on a new computer with Monterey it said the installer needs to be updated. I assume this means the application is 64 bit but the installer in 32 bit. In other words it looks like it can't be resinstalled or moved to a new computer.
I am still using photoshop elements 11. I had been using photoshop previously but abandoned it when Adobe went to the subscription model. By and large I find my old version of elements to be quite useful and I've felt little motivation to upgrade to any of the newer versions. It's not mentioned in the review but there are some downsides to upgrading. Newer versions require setting up an account on Adobe's cloud (doesn't appeal to me). It is also my understanding you cannot open the editor module directly (something I can do with my old version). I don't care for the organization features.
Personally, I wish Adobe chose a different path for photoshop elements upgrades. I would love to see enhancements to the basics, e.g., more refined masking and adjustments. Most of the AI stuff seems like gimmicks to me. Certainly something I wouldn't use frequently. To quote from the movie Amadeus, "Too many notes"
Adobe Photoshop Elements Version: 15.0