I think the point is psychology isn't perhaps solely science but more interdisciplinary. Ultimately if we define science as truly *hard* science, namely, things you can prove using tests about the physical/material world, then many things we hold useful today aren't science.
The closest approximation to science when we care about concepts that don't correspond one to one with physical data is when we measure other kinds of data -- that kind of data-oriented analysis is what led to the Five Factor Model and such tools.
These are NOT strict physical science. They are quantitatively very well-tested tools though, meaning they are measuring something that's not nonsense....but does not necessarily correspond to a material object in full.
However, they are sufficiently data-driven and thus actually we do see legitimate science able to get at their insights -- there's lots of research on the Big 5 and actual neuroscience. But the point is we should not aim to REDUCE psychology/personality to neuroscience. It interplays with philosophy for instance. How people see things and make meaning out of them is often rigorously conceptualized in philosophy.
Jung was very philosophical, which is why some of his types theory is appealing. But he was a bit loose on science, so it takes some work to resolve his insights from a bunch of interesting ideas to something that can be used to really study types of people.