No. In fact everything is executed locally in your webbrowser. No data is transferred to the server at all.
How can I update the birthday calendar?
Whenever you have new contacts you can simply redo these steps. Every birthday event has a unique ID. Means existing events in your calendar will not be duplicated when reimporting them again.
Can I do this from my mobile phone?
It is not recommended to do the steps from your mobile phone as some steps may not work as expected.
But here some guidance:
Open the facebook link in your webbrowser, not in the facebook app.
Typcially you will be redirected to the mobile version of the facebook website. Request the desktop version via the browser seetings.
Storing the website and uploading it here should work.
At least the mobile app of Google Calendar does not allow to import an iCalendar file. You have to do it from your PC. Other calendar apps may support this feature.
Please visit the github project of this site for any questions or remarks.