I figured out how to fix it.
So its basically like this - if you lose physical access to your phone, then ALL of registered websites etc in Authenticator will be gone.
If you have Google Auth backup codes - you can use them ONLY to login to google, and there, to set up Google Auth on new device. BUT IT WILL NOT HAVE ALL THE SITES YOU HAVE USED AUTH PREVIOUSELY :(
So to fix things up you need to contact / manage ALL of those websites/services, each separate, one by one.
Some will allow to disable 2-point auth with alternate login confirmation (SMS, phone call)
Some - like Plarium - will not.
With Plarium and similar ones you need to contact support directly (i made ticket, they responded after 10 hours..) or use PLARIUM BACKUP codes, as they would be from google auth.
So to login - you take first backup code. Then you go to settings and disable google auth - again it ask for code from google auth, so you take second code from your plarium backup codes.
They are (as almost every backup code) one-time use, so count them. You have 6 of them by defaul.. if you have written them to file or printed and stored :)
Hope it helps, I have fixed my plarium with these codes.
But FYI - support responded and was eager to help => they would ask you for proofs of last payments ingame, your plarium account first registration time and place