Sure. You have to store somewhere in your client app in order to use it multiple times. Suitable place should be the local storage or a cookie. It can also be stored in the session storage. The original question was about to store the result in the db while registering. So my answer related to the generation part on the auth. server.
I think its better to clarify what JWT token is. It is nothing more than a document issued by an authority. Same as your personal passport, driver license or ID card. Therefore it basically contains your claims but theoretically can contain any other information.
The key point is it can be verified by a third party without contacting the original issuer. Its a certificate. Same as when you have to show your ID or passport to identify yourself in the real life. Your ID contains several unique details and others can easily check if it was fake or real.
So JWT tokens are virtual ID cards. Issued by an authentication server as the authority. After you have the token any other service which need to identify you as valid user must decide if that token was fake or real on its own.
I recommend to read this brief introduction about the technical details: jwt.io/introduction
So my answers to your questions: