- Feynman was younger than 15 when he took it, and very near this factoid in Gleick's bio, he recounts Feynman asking about very basic algebra (2^x=4) and wondering why anything found it hard
- the IQ is mentioned immediately before the section on 'grammar school', or middle school, implying that the 'school IQ test' was done well before he entered high school, putting him at much younger than 15. (15 is important because Feynman had mastered calculus by age 15, Gleick says, so he wouldn't be asking his father why algebra is useful at age >15.)
- Given that Feynman was born in 1918, this implies the IQ test was done around 1930 or earlier. Given that it was done by the New York City school district, this implies also that it was one of the 'ratio' based IQ tests - utterly outdated and incorrect by modern standards.
- Finally, it's well known that IQ tests are very unreliable in childhood; kids can easily bounce around compared to their stable adult scores.
So, it was a bad test, which even under ideal circumstances is unreliable & prone to error, and administered in a mass fashion and likely not by a genuine psychometrician.
As the saying goes, the plural of anecdote is not data, and this isn't even a very good anecdote. (I charitably assume that Feynman isn't joking here about the score; Gleick gives no source.)