We get this message on one of our machines.
The license server is the newest (2021.1)
Same message from different user only on one machine.
The only different for this machine is that Pro was installed originally on 😧 , then it was uninstalled and reinstalled on C: - I have no idea on how this should affect in any way.
Did you try rebooting the ArcGIS Pro machine, and give it another try?
An alternate option would be a clean uninstall of ArcGIS Pro (remove folders including the ones in AppData folder, clear registry,etc.). Reboot and reinstall ArcGIS Pro again.
I would also suggest running the LSA installer as an administrator, to ensure that the latest version installs for the entire machine, not simply one user.
Just last week, we attempted to upgrade our LSA to 2021.1, but because the user running the installer didn't have the appropriate permissions, it didn't actually upgrade the LSA instance our licenses were on, but rather installed a new instance just for their user. Re-running the installer as an administrator solved the issue for us.