Is mentioning submission rejections in SOP wise? I kind of want to mention in my SOP that I have been through rejections many times so as to hint the reader that I have a more mature mentality to persevere when I am a professional researcher. But, a question that naturally follows is: Would this act instead backfire?
I doubt the signal you will convey is that you persevere through your mature mentality but rather that you may lack the skills to perform science or write acceptable manuscripts of sufficient quality. This said without intending anything personal since only you know the reasons for the rejections. 

Given that rejection rates in journals are generally high, 50% is not uncommon and higher rates exist, everyone has papers rejected for one or another reason. That said, however, repeated rejections signals something is wrong, in the most positive light it could mean your research is novel and have difficulties being accepted by the community (the problem is then the community or how you try to communicate) but as stated first, the most common reason is that there is some deficiency in either the research or the reporting (most often) of the research and this is most likely what people will pick up upon.

As a final note, rejections can be something positive if you learn from them to improve later manuscripts. To use them as a learning experience is therefore the best way to use them.