I think MBTI is a descriptive system, not a predictive one. For this reason I don't think it's like either astrology or science. For example, I might classify people as 'tall' and 'short', and it doesn't make sense to ask whether that is scientific. I make no predictions from my system, I just decide on a semi-arbitrary classification of other people. We could debate where I draw the line between tall and short, but the terminology itself is simply to make discussion easier. If I said Leos were taller than Virgos, though, that would be predictive, and subject to criticism.
Let me take a factor which makes up part of MBTI: extroversion vs introversion. Does it make sense to use this as a binary classification? I think it does - like left and right handedness, we see some behavioural clustering. Some people are noisy, risk taking, sociable, garrulous, easily amused, easily bored etc. These behavioural traits cluster, and we call that extroversion. I think the only way we could say the classification was invalid would be if behaviour was so chaotic or diverse that we couldn't find that regularity. But I do find that regularity, and so I think the terminology is valid. But only valid as a way of making it easier to talk about how people are.
A more severe question about this classification I think is the issue of normal distribution - the so-called 'bell curve'. Many human characteristics have 'normal' distribution (hence the name I guess). Height is normally distributed in a population: extremes are rare, and most people fall in the mid-range between very tall and very short. If introversion/extroversion is normally distributed, most people are somewhere between the two, and putting everyone into one bucket or another is a crude system.
On the other hand, many human characteristics are not 'normally' distributed. Left and Right handedness is more usual than being ambidextrous. Male and female are more common than hermaphrodite. So there are two types of human diversity, the binary cluster and the centralised cluster. MBTI assumes a binary clustering of certain personality traits.
MBTI makes no commitment to personality being innate or learned, and I think it is possible that the pressure of the environment forces people to diversify into clusters away from a normal distribution. If you are a bit introverted, the discomfort you will face in social situations might make you more likely to avoid such circumstances, which means you get out of practice, and so on. This is just an example of how experience might act on a slight, and perhaps normally distributed, innate preference, and turn it into a powerful binary split in society.
I might talk in another post how I think personality typing helps to make people appreciate each other more, which is its attraction for me. This post is just about its intellectual roots.