For a wallet to be and remain cold on a Tails persistent partition, that Tails would need to remain offline as long as the wallet is stored in Tails. Such a cold wallet could only interact with an online watch-only version of it via PSBT files. This is more tediuous than using a hardware wallet where most of them don't have any network communication ability (they interact with an online piece of wallet software in a limited and secured way, never revealing private keys to the online software, so that such a hardwallet is effectively cold).

If you use Tails with an encrypted persistent partition, do not forget to properly document your partition encryption passphrase, multiple redundant backup copies are recommended. Also don't forget to backup your wallet's mnemonic recovery words with redundant copies. You should never loose all details required to restore your wallet properly.

I highly recommend to verify such a restore properly and safely. Only if you succeed with restore, you know you have a working backup. Don't fool yourself, precious coins are at stake.