Very deep questions! The idea of archetypes comes from psychotherapy, in particular Carl Jung. For psychology being a very young science, this makes Jung's psychology even more so. So it's definitely easy to push the archetypes aside as "pseudoscience," but then you would be missing out on their point and value.

As far as my understanding goes, Jung's "research" is observational, based on his extensive experience as a clinical psychologist. He would see something in a person, and then probe them about it, and try to abstract the process from them to help others with the same method.

He also read a significant amount of alchemical and religious literature from many time periods, which he believed to be projections of the human psyche across history. In essence, his scientific method was statistical analysis-- pointing out a million instances of interesting psychological data and then being like "this can't be a coincidence."

So no, I don't think he's scientific in the materialist reductionist sense. I think he's scientific in the "I am sharing what I saw so that you try it for yourself and see what happens" sense, like ancient science.

We can do physical experiments to gain physical knowledge, like "when I mix these two chemicals, I get this reaction." But you can also do psychological experiments: "When I use my mind in this way, I get this reaction."

Jung's suggestion with the archetypes is: what would your life be like if you treated the psyche and its recurring contents as real?