The developer of this app decided to write code to detect which app installed it and refuses to run if it wasn’t Google Play. This is the developer of the app putting in extra work to make the app refuses to work.
If you can download the APK file from somewhere (like APKMirror, APKPure, but be careful to make sure you don’t download a fake version of the app instead) you can use the following command from your PC to trick the app into thinking it was installed from Google Play (you will need adb installed):
adb install --user 0 -i com.android.vending /path/package.apk
I would also suggest trying to convince the developer that artificially blocking customers from using their app when it’ll work fine technically is not a very customer-friendly thing to do (and personally I would look for a different app as the developer has shown willingness to do extra work to harm customers).
No clue about this one honestly, but I’m using Fennec F-Droid which is a version of Firefox with closed source components removed and it works fine. DuckDuckGo Privacy Browser is also on F-Droid, maybe that version will work better.
Hope these notes help you.