For sharing only between Apple devices, by far the easiest method is to put the database files on iCloud and download the Strongbox app for iPhone. That works flawlessly and is easy to set up.
To add non-Apple devices to the sharing group, Mike Cowper's answer is probably the easiest (use Dropbox, OneDrive, or Google Drive instead of iCloud). But I'll mention one quite technical solution that is explained here; it essentially uses three Git repositories: one on iCloud, one on Github (or some other host), and one on your Linux VM. The repository on the iPhone is controlled by an excellent app called Working Copy, which costs $25. You have to push and pull changes for them to sync, and some people won't like taking the extra trouble.
The article is about syncing Obsidian files (a note-taking program), but everything is exactly the same for any type of files, and the method lets you keep your database on iCloud so you don't lose the smooth, trouble-free sharing across your Apple devices. To add another non-Apple machine, you would just set up a git repo on it and link to the Github repo.