I want to share birthdays of people in my contacts list with my spouse and show them in our mutual calendars. I primarily use Contacts on the Mac, but Contacts replicates via iCloud to my iPad and iPhone. This is an aggravating concern, as I keep the Grandchildren's and other important Birthdays on my Mac Calendar and she cannot see these dates.
I was trying to do the same thing. I appreciate your answer, and this seems like a design choice that could be offered by Apple. When I decide to share my calendar, most users will assume (as I did) that everything on my calendar will be shared (why would it not be?).
If Apple feels that this should be a user-decision, then it could offer the option when you share your calendar, do you want to share birthdays (yes or no)?
I would hope that Apple will include this capability in the future. Again, when one shares their calendar, the default expectation is to share all of their calendar (unless given the option to exclude certain items on the calendar).
Unfortunately the options are limited. Birthdays is a special calendar in the Other section that draws its content from Contacts and it cannot be shared.
One option I see is to export its content (from File menu), then create a new Calendar in the iCloud section, import the content you just saved, and share that calendar. You can hide that calendar so you don't see duplicates.
The disadvantage of course is that it's not updated automatically as you add or remove birthdays in Contacts, so you'd have to do a periodic manual refresh.
How to share Birthdays from My Calendar that come from My Contacts with others.
