How to pass a value validated in Webform to the User's profile page?
Using the "Webform Rules" module, you should be able to have the Rules module react on an event like "Webform submitted". So create a custom rule using this Rules Event, and then add the required Rules Actions. Here is a blueprint of these Rules Actions:


Fetch Entity by id, of type user, using the uid of the user who submitted the form (= current user).
Set a data value, for the field of the user's profile where you want to save that birthday, where the value is the birthday entered in the form.
Possibly (not 100% sure) you may also want to add Save entity, to actually save that field in the profile.


Make it right before you make it better ...

But what about a user submitting the same form again, with a different value for birthday? If you don't want that to happen, you might want to add the Flag module like so:


Flag a user after submitting the form (using Rules), using some global flag.
Improve the 1st rule above, to add a Rules Condition to verify that the user has not been flagged yet.


PS: This answer applies to D7 ...