The Tuesday Birthday problem is related to you not having info about the order leading to extra possibilities that you don't intuitively consider. Like in the simple example, if one son is a boy, probability of other being a boy is 1/3rd because the info you have the sets available could be BB, BG or GB. But if the younger son is a boy, then the GB possibility is removed, so it's 1/2 on the older son - BB or BG.
In your case, your younger kid is a son born on xDay. If you assume you will have another kid, chances on being a son are still 50%. It only gets interesting if either kid could have been a son.
In Monty Hall, the info that you don't intuitively consider is that Monty is providing additional information. You think of it as 50/50 because Monty ruled out a goat and car has to be behind one of two remaining doors, but actually the way it works is you chose a door that 1/3rd had a car, leaving 2/3rds chance of car on the other two doors. Then Monty eliminated one of those two doors, still leaving 2/3rds chance of car on the remaining door. So you should switch to that door. In Monty Hall, first you divide the set into a 1/3rd chance group of 1 door and a 2/3rd chance group of 2 doors, then Monty makes the second group a 2/3rd chance group of just 1 door.
In your case, your younger kid is a son born on xDay. If you assume you will have another kid, chances on being a son are still 50%. It only gets interesting if either kid could have been a son.
In Monty Hall, the info that you don't intuitively consider is that Monty is providing additional information. You think of it as 50/50 because Monty ruled out a goat and car has to be behind one of two remaining doors, but actually the way it works is you chose a door that 1/3rd had a car, leaving 2/3rds chance of car on the other two doors. Then Monty eliminated one of those two doors, still leaving 2/3rds chance of car on the remaining door. So you should switch to that door. In Monty Hall, first you divide the set into a 1/3rd chance group of 1 door and a 2/3rd chance group of 2 doors, then Monty makes the second group a 2/3rd chance group of just 1 door.