I was bored and started looking around in the Lessons menu. I clicked on the pregnancy lesson, and at the end it stated "Your Sim will receive a Birth Certificate that you can place on the wall!" I...
If I remember correctly, giving birth at home or sending the sim alone to the hospital were base game features. That's why you don't get a certificate if you chose those options. But actually joining the pregnant sim at the hospital came with Get to Work and that is the only way to get the certificate. Birth certificates were introduced with Get to Work.