I'm about to tear my hair out on the same issue, but finally figured out how easy it is to work on this Google calendar, either on a PC or on an Android phone.
On an Android phone: create an event, such as "Fido Birthday". Then go to this event and click to open editing. Under the date and time, you will see "Not Repeatable." Click this. You can change it to be repeated annually. Save, and ... Voila!
In Google Calendar on PC, the same thing. Create a birthday as a one-day event, then come back to edit it. Right below the date and time (top left) you will see the checkboxes for “all day” or “Repeat”. Check the "Retry" box and select "Annually" from the drop-down menu. Save, and you're done.
Your dog’s birthday is now on your Google Calendar. :-)
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