What if I generate token on my side and fill it with data I saw, how the system will distinguish my token from it's own ones?
Sure. If you know the secret the server uses, you can do this. If the secret is not secret anymore, the security model of JWT Tokens are fundamentally broken, and nothing can rescue it.
Typically HMAC SHA256 is used for authentication:
JSON Web Token (JWT) is a compact, URL-safe means of representing claims to be transferred between two parties. The claims in a JWT are encoded as a JSON object that is used as the payload of a JSON Web Signature (JWS) structure or as the plaintext of a JSON Web Encryption (JWE) structure, enabling the claims to be digitally signed or integrity protected with a Message Authentication Code (MAC) and/or encrypted.