I totally forget to check out the birthday voice lines...

No, you don't have to talk to your servants in My Room to unlock birthday line in their profiles

First of all, happy late birthday!

https://www.youtube.com/watch?v=4AcEnKwwmhg

You can check them all here. It even includes the servants that are currently only in JP. Also, you can view them by checking a servants's voice lines when you open their profile. You know when you tap and hold a servant so that you can see their info, sprites, and so on? There's a tab for voice lines. I hope this helps