I'm from Kenya.
In this modern age, there's absolutely no way a birth would be recorded 1 year or years later.
An hour or 2 (for those countries that time is listed on BC), I see that happening. But years as was said by OP? Definitely a NO.
Even for 1 day mishaps, I don't see that happening. Convince me please, that you didn't note that your kid was born on a certain date. That even the mother doesn't know?
Even births at villages are recorded by midwives and they go to hospitals same day or chief's to get it official. At birth there's: midwife, the parents, relatives and few friends. There's no way one of them won't remember the actual day the kid was born. Additionally, it's the midwives (in the villages) responsibility. So, as you can see, even in worst areas (villages), the date is recorded.
Sure OP's kid was born in a better environment eg hospital, clinic etc.
P.S: I think for births in some areas pre-1970 (not sure of exact year), USCIS/ DoS acknowledges that records may be obscured. But post 2000????NO