I think that's a limitation of the Files API. In the least, I've yet to come across an app that allows syncing with third-party storage apps through Files. I've only seen one-way saving or using the file picker to retrieve a file.
"KeePassium can work both offline and with your favorite cloud provider.
There is no setup: pick your database through the familiar Files interface — and that's it. By integrating with the Files app, KeePassium supports almost all storage providers, including..."
Interesting. I don't have a chance to mess with the app, but I wonder if it works because it's syncing individual kdbx files that the user gives access to one at a time rather than a whole directory (including subdirectories). Logseq and Obsidian, for example, would require syncing an entire directory along with subdirectories. My assumption is that they work with iCloud because they're given their own app-specific folder to do as they wish.
If that is the limitation, I wonder if they could get around that by using some type of bundle or archive. Regardless, thanks for the additional information.