My App on the airdrop button says "Register". After taping on it there is a message displayed "Temporarily does not support your device". Does it mean that I'm dropped out of the Airdrop because of my phone?

That would not be fair. It's on Android 7.1.1 phone, so it's not really old.
"Temporarily" is a delicate way of saying "it's not supported and maybe never will be".

I have exact the same problem. Pickup button was inactive - taping didn't work. I've reinstalled the App and now i see "Register" on the button like on Comod's screenshot.

Waiting for a fix.