There are still a lot of questions surrounding any kind of trait theory and their exactness. But there are a massive number of assumptions people make when we even start with "science" and "fact." It's impossible to completely unravel it all at this point. I think psychology in its whole has been called a pseudoscience and probably not wrongly.
That aside, Jung observed very basic things and made a scale between the things he saw. There isn't a need for "proof" if you understand the very basics of it.
I'm going to assume the question here is mostly on the functions, not the expressions. The expressions are Extroverted/Introverted and Judging/Perceiving. Expressions are so obvious, there isn't much reason for explain them in the first place, IMO.
I'm approaching this as the very original meaning Jung appeared to have for S/N and F/T.
Jung thought that two people could see the same thing and have completely different interpretations of what they were seeing or understanding.
If this assumption is true, then Sensing verses Intuition is one scale conjured in order to measure this perceiving function.
Supposedly when a Sensor perceives something it is in a logical way. They observe something in the present and in connection to things that are logically connected to it. Sensors don't tend to connect other information into what they sense or see. Particularly T types, if something is broken, it must be fixed, if something is not broken it does not need to be fixed. I have to pair T with this example, because T/F is the deciding factor.
Intuitives don't observe something logically. That means that a number of other factors could come to their minds in the observation of something. Time and place does not necessarily separate instances or items, they are overall inputters and thinkers. Just because it's broken, doesn't mean it has to be fixed to be the same as it was before, just because something isn't broken doesn't mean that it doesn't need to be changed.
Ses experience singular, logical events. Ns experience universal, connective events.
Then there is Feeling and Thinking. These functions were originally based on decision modes. Can people make decisions? Given the same information can they make different decisions? If they can then this is a scale to supposedly model two types of decision making.
Feeler supposedly make decisions based on their gut feelings. Fs usually make connective, universal decisions, which apparently lends itself to people-involved situations. Fs come off more instinctive I suppose. It's probably better described somewhere else. On Wiki it's described as being decision making as if from the inside.
Thinkers are the supposedly logical decision makers. It's often given too much merit, because of our current societies continued 1800s fascination with science and objectivity. Thinkers view a decision as if outside it and hence are more detached.
We probably all know that there are ways people intake sensations and make decisions. What those are, I suppose, could be called into question. Simply, that seems to me what Jung did: he thought or observed these functions.
The MBTI is still frequently used, especially within work and business I've found. The MBTI's purpose was within business if you look up the history.
The question becomes, what kind of proof is everyone looking for? What sort of proof for or against any trait theory? If we don't know then there's no way to approach the question. If the question is whether or not a person can accurately test themselves, then we also have the question if others can accurate test them. Who is in the position to accurately test anyone?
Society has people chronically over diagnosed. I was reading today that one in four Americans have a mental illness. What kind of fact is that? What have we made into mental illness? Soon I'm going to read "Crazy Like Us" which I'm very interested in seeing the verdict in. Historically societies have made ridiculous illnesses out of natural behavior, at least that is what we think of the past. The problem with this is that societies have personalities, They can be defined and compared. We can create stereotypes and characters out of nations the same way we can with people.
We don't have utopian societies, we don't have static societies. There's nothing absolute and static about culture, it evolves. All of psychology is flawed based on this, surprising, so are some more static sciences it would seem.
Pseudoscience is more popular and well known than "science." Today's 100% true 100 facts, four generations down the line will be 10 facts and 90 myths from the early 21st century.
IMO, it is pretty funny.