I think though that part of it is that I have yet to come across someone who has boasted about how brilliant their dc is, whose dc is anything above bright but not exceptionally so.
At the younger age, there is an aspect that from that tiny baby who did nothing, who suddenly says a sentence less than 2 years later, it can seem amazing... as their parent.
Most young "genii" I've come across go into that category. That parents are just amazed with what they can do.
Then add the ones that are heavily interested in one thing, and so can come out with information that seems amazing for their years. If you spend the time talking about numbers and giving great enthusiasm every time they count, they'll learn it quickly.
Ds was interested in planes from an early age. I believe when his form went for a school trip round Brooklands the volunteer who was meant to be taking his group round, ended up letting him do most of the talking, because they said he knew as much as him-who had volunteered there for 15 years. Take a child and give them an interest and they can absorb knowledge like a sponge. Doesn't make them a genius. Just wish ds was as interested in... well almost anything he does at school.
On the words front I remember someone I knew wanting to put a complaint in about their health visitor. They went for a 2 year check, and the HV asked about their dc "do they say 50 words?"
She replied "I think he's going to be a genius, he says nearly 120!"
Apparently the midwife didn't recognise the amazingness of this.
You also get people who are comparing how they did with their dc's ability. Dm is a maths teacher and she met someone one time while waiting for an appointment who, after finding that out, spent the next half an hour raving about how their dc, secondary school age, was a genius at maths, which was fantastic as they'd always struggled. Dm nodded and listened without saying much. As the other lady was called she lent over to dm and said in confidential tones. "I don't like to boast too much, but I knew he was brilliant at maths when I found he'd got into the third set. And there are eight sets! They think he'll get a C!"
If you've always found something hard, you admire it in others. I think that increases when it's your dc.