This is reminiscent of social media's favourite, the Dunning-Kruger effect, where in order to assess the level of an attribute that applies to them, an individual must use that attribute itself.
This suggests that there's no relationship between self-assessed and measured intelligence at all as opposed to Dunning-Kruger's exaggerated gap at the low end.
I wonder why Dunning-Kruger doesn't seem to apply here.