>I mean where am I supposed to store the recovery key if not in my password manager?

My password manager (keepassxc) supports multiple databases. I store all TOTP recovery codes in a separate database (and with a different unlocking password) from the one that has regular passwords and the TOTP secrets.

All my databases are backed up on someone-else's-computer, but I only keep the regular-passwords-and-TOTP-secrets database synced to my devices.