I read that uuid does not bring any security advantages
This entirely relative to a given context. So it's neither true or false.
Consider that right now the session id is encrypting the auto-increment id (no uuid is used). If someone manages to know how the session is encrypted, then he can impersonate all the users: encrypt "1" and set the value as sessionID, encrypts "2" and set the value as sessionID, etc.
Session identifiers work if they're long random pieces of information. They do not encode or encrypt any information, these tokens are used by the server to locate information pertaining the established session.
In a typical scenario, client A connects to server B for the first time. They have no information or session id at this point. Server generates a new session id and sends it to client. Potentially authentication occurs and some data is stored on the server pertaining that particular session. Every subsequent request from the client carries this identifier so that the server can match the data relevant to that particular client during this particular session. Notice the data is stored on the server, all the client does is issue requests of whatever kind and tack on the session identifier as a way to maintain state in a stateless system.
Simultaneously other clients are doing the same. The server can maintain multiple states since every client uses their own unique session identifier. If the sessions weren't random or easily guessable, then an attacker could calculate or guess them and hijack established sessions.
So a randomly generated UUID is no better or worse than a randomly generated session identifier for the same length of random data.