Birthday Adapter creates its own calendar account 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.
Check that your calendar app shows the birthday adapter. For google calendar you need to go to settings > manage accounts and activate the birthday adapter. In FossifyCalendar you find this setting at CalDAV > manage synchronized calendars
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.
This is done once every 5 days and invoked by Android itself together with other synchronizations.
Please check that permissions of the app do not expire. Also make sure to allow background activity for the app and to set it to 'not optimized' or exclude it from battery optimization
Please go to hosted.weblate.org/engage/birthday-adapter/.
There is an issue tracker on github.com/SufficientlySecure/birthday-calendar.