From what I saw on GitHub (it’s where bugs are reported), I think unfortunately you’re out of luck as your MacOS version is way too old – and out of support.
It appears that the Mac App store disagrees with the Bitwarden download page (and GitHub as @nail1684 noted). Guessing that someone ought to update the App Store’s information.
@Aerozolic After checking in, Bitwarden is supported on different macOS versions depending on if you download from the Bitwarden website or from the app store, so be sure to drop a bug report or contact support if you’re still experiencing issues.
The problem is with the extension. The only way to get Safari extension is via App Store? If so then this is not working in Safari 17.6. Don’t care much if the app is supported or not, just the extension.