@Lanski nailed! When clicking this “backup now” button you download the /root/.config/ethereum2 folder which contains all your private keys + passwords + account metadata (handled by ethdo)
To restore you can just go the same UI the @Lanski showed, click “Restore” and select the file that was backed-up originally.
And yes, the ideal would be to have a uniform solution across clients. Using the same validator keys across multiple networks is not recommended. You should do deposits for different keys on each test network do you want to participate.