Internet Identity does not directly support sign in flows between different browsers (e.g. Telegram browser and Chrome browser).
Besides Telegram mini apps, this also affects the development of sign in with Internet Identity in mobile apps.
So instead the recommended approach is to implement a flow where you authenticate with Internet Identity in the same browser to a website on behalf of your app. Then this website can delegate towards the app with e.g. universal link.
More information can be found in this forum post and the linked documentation within that post: New security best practice on integrating II for mobile apps
I have no experience with Telegram mini apps, not sure if it’s possible to link to a browser outside the mini app and then link back to the mini app. Any insights and/or feedback in this regard would be highly appreciated.