Birthday Adapter creates a new SyncAdapter on your Android device, which synchronizes all events, like birthdays, anniversaries and so on, from your contacts into a calendar. This calendar is displayed in your standard Android Calendar application.
This is not the idea of this app. As you already have all birthdays in your contacts and synchronizing these into the cloud, it would be redundant to also synchronize a calendar with the same information. Instead use tools on your desktop pc to display the birthdays from your contacts in your calendar, like Birthday Adapter does. In Thunderbird this can be done with the Thunderbirthday Add-on.
Please report it in the bug tracker of Birthday Adapter.