Hi @Anonymous,
There is currently no similar trigger for Office 365 and Azure AD like you mentioned.
If you want such a trigger to be supported by MS Flow, you could post your idea in the Flow Ideas forum, so it might be considered for future releases.
https://powerusers.microsoft.com/t5/Flow-Ideas/idb-p/FlowIdeas
In addition to the two examples you mentioned, perhaps you could achieve your needs through other means.
Take the example of birthday:
- Use Recurrence as a trigger to run Flow once a day.
- Configure Search for users to get all user ids, then configure Get user profile action to get user's birthday. (Default everyone has set their own birthday)
- Configure Condition to determine whether Birthday is equal to today. If yes, send the email with the user's information and blessings.
- Another thing worth considering is that the same day may be the birthday of multiple users, so you could use variable to store the information of these users, send only one message containing these users.
Image reference:
Edit:
Considering that there may be no birthday on that day, we need to add a condition to determine whether the variable is empty. If it is empty, no mail will be sent.

Expression:
@equals(formatDateTime(body('Get_user_profile_(V2)')?['birthday'], 'yyyy-MM-dd'), utcNow('yyyy-MM-dd'))

Please take a try if it meets your needs a little bit.
Best Regards,