Why...? I don't know anyone who actually cares all that much about royal births. If a woman had a child the same day... unless she's only got really crappy people in her life who don't care about her, I'm pretty sure most people will care more about the woman and baby they actually KNOW.
Everyone shares a birth date with many other people, and some of those people will be born into wealth, fame, etc, or find it in life, while many others born on the same day will not. It's just how it works, it doesn't make someone better for having been in a different womb.