There are two possibilities to show birthdays/anniversaries of your contacts in your Android calendar:
- birthday events are generated by the server (for instance, this feature is offered by Nextcloud),
- birthday events are provided by a local backend.
Local birthday backends
By default, Android does not maintain a calendar which contains the birthdays of your contacts. Some device manufacturers have implemented that in their firmwares.
You can also use these apps, which will read your local contacts (so they need contact permissions) to maintain a local birthday calendar: