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I hesitate to use the same app for both authentication factors.

The reason why I started using Authy a long time ago is that it supports multiple devices and isn't linked to any other account (such as Google or Microsoft).

Also KeePassXC -- if you don't like the idea of 2FA codes being in the same db as passwords, it's straightforward to use a separate db for 2FA only.

Manage your own sync between devices with syncthing, dropbox or whatever you prefer.

Personally I dislike the idea of putting the other factor(TOTP) alongside the main two ones (email/password). Kind of ruins most of the purpose of TOTP and MFA in general.