But if an attacker gets your fake birthday and uses that to successfully reset credentials on another site that uses the same fake birthday?

At some point it becomes your birthday of record as far as the internet is concerned. Doesn’t matter what the actual record says.

I use the 1st of my birth month. Slightly less suspicious? It's at least a little easier to remember. Generate fake profiles and identities usually is easier when you have bits that are rooted in your actual reality. Like, you have the same zodiac sign either way in this case, so you don't have to remember two of everything. Or if you're talking about a birthday trip, or related birthday thing from the past, details about the weather would be consistent, etc.

I heard from a number of Syrian refugees that this is actually very common in countries like theirs, where births may not be recorded, records are lost or destroyed. Some people don't even know their exact date of birth and they would typically enter January 1st on forms like this too.

Same, my D.O.B. is 1/1/1970 for anything that doesn't justify having the real deal.