Yep, happened to me too. I thought it was some kind of a browser issue. I guess I was wrong...

Well, since you love KeePassDX as me, I started to use KeePassDX as a "database of notes". Under Root, you can create categories by adding 'Group's. Inside each group, you can add an 'Entry' (a note). each Note, will have a sub Category (Title), subject of note (User name), url and the note itself (note). you can also add tags to your notes.
Give it a go, you'll be surprised how well this works, finding/searching for whatever is there.

You could install Droid-ify, it has a setting that lets it automatically install F-Droid apps

What about Collabora Office?

Same here

Should this work on an Android device (I am using a brave browser), or just on desktop?

It gives me a GPT3.5turbo and GPT4 options. Not sure what's "turbo".

I am just a regular 3.5 user

I've added an API Key and still getting: An error occurred while fetching a response.

Another vote for Bitwarden

OK, I see what it's doing. If an app was removed from the favourite list, the import of the backup file will add the missing app back to the favourite list.

That actually does not do what I was after.

I am still looking for an app that will recognise all the installed F-Droid apps, so that I'll be able to backup the apps to a backup list. Then use the app and the backup list to reinstall the apps on another phone.

I was able to use Neo Store to export favourite apps as you suggested, but I wasn't able to import apps back, not sure what am I missing ....

I use Simple - Social Browser - https://play.google.com/store/apps/details?id=com.creativetrends.simple.app

Thank you for your comments.

Neo Store, does show you all F-Droid apps and lets you export all favourites apps to an .xts file. I uninstalled few apps from my phone, but I wasn't able to import the "missing" apps from the list in the .xts file.

APK Kit lists all instated app on my phone, including Google play store and F-Droid, so this is not very useful (to me).

As for NeoBackup, I do need an app for a non rooted device

and Restic, is not that clear how to backup list of F-Droid installed apps and how to use it to restore missing apps.

Great list, here are some more apps to consider to add to the list: InnerTune, Amaze, DiskUsage, DuckDuckGo, K-9 Mail, KDE Connect, Video Transcoder, Notally.