This is NOT the Google enabled Authenticator. This is https://f-droid.org/en/packages/net.bierbaumer.otp_authenticator/
There is no signing in.
Is there anyway to move credentials / accounts over to another phone? At worst, it seems like you need to just re-add them to your new phone, but maybe I'm missing something.
The app doesn't offer any feature to do this, as you can see from the issue log. That leaves with two choices
Using backup methods for only an app as discussed here Full Backup of non-rooted devices
If your device is already rooted, using titanium-backup is a popular choice (amongst other such apps)
But I suspect your device is not rooted, so you have only one choice
Q & A