It’s private data so even if a user wanted to expose that information Twitch may not want to for privacy/legal reasons.
Even if Twitch did allow users to display that information, there are a dozen ‘recommend’ connections, and an unlimited number of apps that can be connected to an account, all of which may have overlaps in usernames between different users. For example my XBox gamer tag could be identical to the username of someone else on Twitch, or a connected app on another service.
And even if a user did want to display that, and Twitch did want to allow it to be displayed/searched, it’ll still unlikely be returned in the 3rd party API because developers of apps already know who’s connected, and if you want to see a connection of an app you have nothing to do with then there’ll be the issue of returning partial results, as a user may want to share their xbox connection, but not their Amazon account.
… And even if all of that did happen, searching by xbox gamer tag would still result in a poor chance of successful searches as the majority of users wont go into their settings to toggle on their connection to be shared. When it comes to connecting to apps the vast majority of users are set it and forget it, meaning a large number of xbox gamer tags would never be searchable even if that user is on Twitch.