Manychat will not execute this function because by TOS and messaging policies you’ll be contacting a user outside of the alotted 24hr window in most cases. 

There are two workarounds:

The first:
Run ads specifically targeted to someone’s birthday.
Push this into a Messenger funnel.
Deliver your discount.

Second:
Create your MC funnel and collect the contacts’ names and emails to Google Sheets.
Use IFTTT or Zapier to then take those names and fill your email list (have a segmentation or list setup specifically for this import); or fulfill this step manually by yourself or VA
Set up a birthday flow in your email client that will send that coupon or discount the week of or on their birthday with an additional follow up reminder in case they hadnt taken action

This would be my personal approach. I use birthday funnels for my audiences for multiple reason. Hope this helps.