Since a few years ago, the most common and traditional method to do that was to store this data on the server using either a memory-based, disk-based, or external session manager. Each session can be retrieved using a unique ID that the client receives with the authentication response, usually inside a session cookie, that will be transmitted to the server on each subsequent request.
Here's a brief diagram showing the Session-Based Authentication Flow:
This is still a very common technique used by most web applications. There's nothing wrong with adopting this approach, as long as we are okay with its widely acknowledged downsides, ...
