If you want to modify text/number in a form then you have to use the styled() command. Check the link for a detailed list of options. You can put it in an if statement to check for if the month and date match.

if month(birthdayfield) = month(today()) and day(birthdayfield) = day(today()) then
styled(text(age(birthdayfield)),"green","red")
else
styled(text(age(birthdayfield)),"","")
end

If you want to use conditional formatting in a table view, then you have to first create a field that contains a number or text. So you can create a formula field and use the age() command to give you a number:

age(birthdayfield)

Then add this field as a column in your table view. Then you can add something like this in the f(x) part of the conditional formatting window:

month(Date) = month(today()) and day(Date) = day(today())

Then change the other settings to your liking.