Nop. I was logged in with another account. I signed out and signed in with the affected old account and I realized that Find My is working but Airtag was not visible at the Items section. So I took out the battery from the Airtag 5 times and paired again with the phone and everything worked well.

For me this issue means that my paired airtag is unusable and I throw out the windows 100$. This is ridiculous that it is only used with one icloud and cannot be reached online through browser only from the phone. Thus you could unpair it.