why do I have strangers’ birthdays on my calendar app?

To me it just sounds like you gave an app permission to access your Contacts and it made a mess of them. What apps do you have listed to allow access to your Contacts at Settings > Privacy & Security > Contacts?

Your Contacts can also be changed by any device that is also logged into your account. Go to Settings > [name on top], and scroll to the bottom to see a list of devices that are currently logged into your account. If there is a device logged into your account that is not one of yours, then you will want tap in it to remove the device, and follow that up by changing your Apple Account password.

• Do you have these people in your Contacts app?
• When a person shares a contact with you, they choose what data they want to share?
• Do the Birthdays appear on a different Calendar when viewing the Calendar app?
• Do you share your Apple Account with anyone else?