Science as we understand it today is a systematic organization that builds and organizes knowledge in the form of hypotheses and testable explanations about the observable world.
It isn't. For the best example: the oldest and most "science" science (physics) has, as one of its greatest achievements, demonstrated that the foundation of reality is unobservable.
Is there too much subjectivity in psychology to make it a formal science?
You treat psychology rather singularly. Psychologists work in fields like Human-Computer Interactions, computational intelligence, neuroscience, cognitive neuropsychology, social psychology, neurology, and a great deal more. I've been an ardent critic of many branches of psychology for some time, but that has nothing to do with the field. The same issues that caused such as stir with the hockey-stick graph are ubiquitous within social and behavioral sciences because it is too easy to feed data into some modelling or statistical package and obtain results (even if they're garbage).
Psychiatry, on the other hand, performed one of the most successful hostile takeovers in business history. When the medical sciences barely recognized psychiatry as science, when insurance companies were pushing psychiatrists for some way of classifying what was a medical illness and what wasn't, and when therapists without medical training were claiming that, as psychiatry wasn't medicine, there was no reason for its privileged position, the DSM III came out of nowhere. Now, all mental disorders are distinct illnesses because psychiatrists said so. It's completely inaccurate, and it has successfully impeded scientific understanding of mental states, disorders, and diseases, but it did keep psychiatry in business.
Are there a slew of studies published by psychologists that are employ bad methods to get corrupted data which is then subjected to improper statistical analysis? Yes. The same is true within psychiatry , economics, the many fields subsumed under climate science, cosmology, the many fields subsumed under sociology, behavioral genetics, and more.
Psychology often lacks the precision and rigorousness of other sciences, such as mathematics or physics
Mathematics is no longer really considered a science. For a very long time it was, and the ultimate goal was to come up with the axioms necessary to derive any and all mathematical statements. That was already ending by the time Gödel sealed the coffin. As for the other sciences, what are they? The two largest areas of research in physics are quantum physics and realistic physics, and they contradict one another. There is as yet no agreed unifying solution to them. While physicists have made tremendous progress dealing with non-biological systems, physicists working within the life sciences have not done the same. We have just recently been able to create full models of simple cells. Ecosystem models not only disagree from one another but what the methodology should be.
The "end of reductionism", which had by the late 19th century at the latest become
the approach to scientific issues, came to a grinding halt because of physics, and it was within physics that the first and most convincing arguments against the 19th century empirical reductionism as an adequate approach in and of itself may be found.
If you are talking only about clinical psychology, then you should be talking about why psychiatry isn't a science.