Visual/physical identification verification
Assuming that there are truly 8 cognitive functions, and everyone is born with one in the dominant position, I think there should be simple (or at least compelling) ways to test for this.
Firstly: physical differences
Surely there is some physical representation of each function in the brain. Especially if everyone is born with one of the 8. There might be visible differences in the brain.
There might also be pointers to functions in DNA itself. Take for example visual identification. If it's true that certain features do map to certain functions, then this points to some connection between the genes for the function and the genes for that feature. They must be near each other when the division of the chromosomes happen.
If you believe that its more about expressions than real physical qualities- okay fine. Still, I think in that case you can train some A.I. model to notice these expressions. But what expressions? Expressions as reactions to a stimulus? In that case it might be possible to show people different images, objects, or sounds and then capture their expressions to those. Is it only their natural expression? Then it becomes more difficult, since you have to consider what natural is. Maybe the expression reactions themselves are different? For example people may smile to some stimulus but the way they smile is different. In that case, you should collect a collage of people showing different emotions and try to map it into 8 different categories.
I'm using 8 instead of 16 because it should be easier to group things in less than more by similarities (as the similarities can be more general). Besides, I think there is some value to how Jung only had 8 long ass entries into each dominant function with a little paragraph or two into auxiliary functions (and nothing into any of the other positions).
I was going to write about using statistics and some other cognitive testing... but I got bored while writing this.
I have a suspicion that these 16 types would follow a general trend in terms of IQ too. Even if IQ is not important, or it has cognitive biases favoring types- that does not rule out evidence that people tend to like and get along with people around the same IQ as them. If the theory of duals are correct, then it'd be quite funny to think of the consequences of this.
difference is their behaviour, including facial expressions/body language
facial expressions/body language are important assets when typing somebody