It is supposed to be a science. It is commonly referred to as a "social science." Psychological researchers employ the scientific method, and in that regard it is most definitely a science. Yet the predictive values of it, in practice, are a lot less reliable than many other sciences would lead us to expect. While many experiments seem replicable, others are much less so.
Looking at it through the lens of art makes a lot more sense to many people. While the study of it is, perhaps, more scientific, the practice of it by psychologists would appear to be far closer to a performance art! I doubt that any therapist in the world would insist that so long as the exact same steps are followed that the exact same results will be achieved.
A lot of people just want to take the whole field and dump it straight into the landfill or the shredder. "Man is too complex," they will cry. If a test divides people into 4 or 12 or 16 types, they will note that the options are too few to be all-encompassing. With enough more options, they will merely note that the variety is too numerous to be useful, while still possibly suffering from the other malady as well.
Regardless of which view you take, there is so much more to psychology than we are going to get to in this exploration. It's not our task. However, there are some parts of psychology that seem to more clearly fit our task than others: Intelligence and Testing, Learning, Memory and Cognition, Motivation and Emotion, Personality, and Sensation and Perception.