I’m really unsure what’s going on here. I was using the beta builds of Dropbox, something I’d turned on months ago to test the new FileProvider approach. When I revealed hidden folders with Command-Shift-period, I could see that both Dropbox and Google Drive had a .Trash folder in their ~/Library/CloudStorage/ locations. Deleting a file or folder with Command-Delete in Dropbox put it into both Dropbox’s .Trash folder and made it appear in the system Trash as well, with a cloud icon.
So all that seems pretty reasonable. What I don’t understand is how your system Trash seems to have somehow become linked with Dropbox. That would imply you could see the full contents of your Trash on dropbox.com and that if you delete a file from a non-Dropbox-managed folder, it’s being sent to a Dropbox-managed Trash folder. Which seems really wrong, as you suggest.
To further confuse matters, I decided to get off the beta builds (build 164.3.7907) to see if I could reproduce what you’re seeing. When I did that, in both Monterey and Ventura, I got version 163.4.5456, which reverted me to the pre-FileProvider approach of just syncing /Users/adam/Dropbox. On both machines, so it’s clearly not required for Ventura.
And worse, while I was writing this, my iMac (Monterey)'s version of Dropbox just updated itself to 164.3.7907 on its own. It still doesn’t seem to be using FileProvider—there’s no Dropbox folder in ~/Library/CloudStorage. Now my M1 MacBook Air has also updated itself to 164.3.7907. So I’m unclear on why they’re both running the same version as the beta they previously used, but now aren’t using FileProvider?
So what version do you have installed? Here’s the current list.