"UnifiedPush is a decentralized push notification system that lets you choose the service you want to use. It’s designed to be privacy-friendly, flexible, and open — making it perfect if you want control over your push notifications."
Currently, Thunderbird for Android relies on Google FCM (Firebase Cloud Messaging) for push notifications. For users who do not have any ties to Google at all (meaning no GApps installed), there are currently no options for push notifications within Thunderbird. UnifiedPush has several implementation apps (such as ntfy, Sunup, et cetera) making it easy to set up push notifications as long as the app itself supports integrating with a push distributor.
This was discussed via a GitHub issue starting back in 2021, but the issue was locked without much explanation (outside of some potentially misinformed talk of needing a proxy server):
https://github.com/thunderbird/thunderbird-android/issues/5165