From Google:
"In non-leap years, some leaplings celebrate their birthday on either February 28 or March 1, while others only observe birthdays on the authentic intercalary date, February 29."

weird

Depends where your from. In the uk your parents are forced to chose either Feb 28th or March 1st.

Over here in good ol' America it's the child's choice every year whether they wanna celebrate it March 1st or Feb. 28th. Like one year they're like "feb 28th sounds good" and another they're like "march 1st. nice."

And in their birth certificate and other legal paperwork? How does a baby chose that?

i wouldnt know, but i had a friend that had a bday on a leap year and she said she gets to choose her bdays

Wouldn't their legal paperwork say the 29th of feburary, 20XX?

Birth certificate yes but not the rest of it. Your parents chose your official birthday but you can celebrate it on any day you chose. When your born your parents decide your official birthday.

oh

makes sense