Is it possible to obtain organization client id and use it to authorize user to access twitch API? Or if not, how does the authorization to services like steam and ubisoft works?
Users don’t connect to an Organisation, they connect to an App. If you follow the Authentication Guide in the docs it’ll walk you through creating an app, and how to use its client ID, client secret, and redirect URI to allow users to connect to your app.
If you’re talking specifically about how to get your Organisation to show in the Recommended Connections, that’s specifically curated on Twitch’s end for those specific companies and is not available to the developer community at large. The vast majority of connections for users will be under either Extension Connections, or Other Connections.