TOTP is a completely offline protocol. Basically you scan a QR code which is a signing key that is saved to your phone. Then all your phone needs is an accurate time source, and periodically signs the current time with that key to produce a 6-digit code.

As such, no, it's not associated with anything from the source account. It is not challenge/response, and you can scan the QR code with 10 different phones and they will all produce the same codes at the same time.