Are you saying that income influences what time of the year you are born?
The paradox doesn’t talk about meeting your birthday sibling, but meeting two people who are birthday siblings.
So by the time the 12 year old has met ~20 people, there is a 50/50 chance that amongst those 20, there’s a birthday pair.
If you randomly choose the ~20 people from the global population then yes, this will be the case (especially after a number of rounds). And yes, I'm aware this is also the definition of the paradox.
But if you choose the people from your vicinity (i.e the people you are actually likely to meet), the chances will vary based on your individual parameters (which defines the number and quality of the sample size).