Since birthdays appear correctly on iCloud.com and on your iPhone 15, the birthday information itself is most likely correct.
The issue is probably limited to the Calendar app on your MacBook Pro.
First, open Calendar on your Mac and make sure the Birthdays calendar is enabled. If the calendar sidebar is hidden, choose View and then Show Calendar List. Look under the iCloud section and confirm that the Birthdays calendar has a checkmark beside it.

If the Birthdays calendar is missing or does not update, check that Contacts synchronization is enabled. Open System Settings, select your Apple Account, choose iCloud, and confirm that Contacts is turned on.

If it is already enabled, try turning Contacts off, choosing Keep on Mac if prompted, waiting about 30 seconds, and then turning Contacts back on. Allow some time for the contacts to synchronize again.
Next, refresh Calendar by opening the Calendar app and selecting View, then Refresh Calendars. After that, quit Calendar completely and reopen it.
If that does not resolve the problem, restarting the Mac in Safe Mode once and then restarting normally may clear cached system data that is affecting Calendar.
AI assisted may have been used for the wording; the technical content reflects my own experience