This is the common misconception. Think of a cookie as a storage 'bucket' on a user's device. It can store up to 4KB in bytes of, well, anything (a string). JWT is format. That's it. It is enciphered and has the ability to store JSON, but otherwise, it's just a format or scheme that allows you to organize how you store your session (or whatever) data. As an example, in my usage I actually store my JWT in a user cookie and all it keeps is the expiration, user id and a few other tidbits I don't want to hit a database for but am comfortable exposing in case the token is cracked.

Right, but then the browser is automatically doing the job for you (saving the cookie and then returning in the next request). That would be so bad! We can't do custom code to acheive the same thing...