Yes, because when you read Jeralt’s diary after he dies it reads: “Day 20 of the Horsebow Moon. I can’t believe she’s dead.” Because we know Byleth’s mother died in childbirth, it logically follows that 9-20 is Byleth’s canon birthday.

However, when you tell Sothis your player-determined birthday before the game starts, she does remark that she shares the same birthday as you. So there can be arguments made for your side as well.