To share cookies between subdomains, just add a dot (.) in front your domain add this env variable:
SESSION_DOMAIN=.domain.com
Note the dot (.) in front of it.
But, Laravel session sends an encrypted, cookie so you would need to replicate the encryption/decryption Laravel uses in the other project.
You could add your session cookie name into the $except property in your EncryptCookies middleware, but I don't recommend it as it would send the session id unencrypted to the browser, which is unsecure.
For replicating the session cookie encryption and decryption you can look at the base EncryptCookies middleware implementation, at your ./vendor/laravel/framework/src/Illuminate/Cookie/Middleware/EncryptCookies.php file. (do not change files inside the ./vendor folder)
Once you manage to decrypt it, then you have a Session ID, how to retrieve session data?
You need to check within the session store code, from the store you chose in your ./config/session.php (or .env file) on how this store uses the Session ID to retrieve its content.
And of course, this store mechanism should be accessible from the non-Laravel project.
A bit of work, but doable.