1Password covered this very well:
https://blog.1password.com/1password-2fa-passwords-codes-tog...
https://blog.1password.com/1password-2fa-passwords-codes-tog...
I don’t use a KeePass or similar device, and I do have my TOTP tokens on my phone (using the great OTP Auth app, no affiliation), but I do most of my work on a computer so I toyed with completely removing password management from my phone and just having the OTP app there. I haven’t committed to that that yet though. Another option is to use a cheap device such as an iPod Touch for OTP, though those are probably more expensive than a dedicated OTP device (I just happen to have one lying around unused).