How do people peer-review many papers? I am an Assistant Professor of Chemistry. I try to be an active peer-reviewer but on average it takes a few hours to review a manuscript (reading, some searches in the literature, checking some relevant reference, and finally writing the report).

I review 2-3 papers per month and I need to spend some time outside my working hours. In various blogs, articles, websites like publons.com, I see it is not uncommon to review 10-20 papers per month.

Then, I came to the conclusion that I am doing something wrong. Because by my calculations, it is impossible to peer-review many papers along with various tasks of an academic.
I don't think people who review 10–20 papers per month are likely to be doing a good job.  They are probably just reading a paper once and writing a couple paragraphs (at most) based on superficial impressions.  I've certainly gotten reports on my own papers that seemed to be produced that way; and they were both frustrating and not very useful.
I personally review about 3–6 papers each month, and that is by far the most of anyone in my department; I have also received outstanding referee rewards from multiple journals, mostly just recognizing my high levels of work.  (This also represents a big part of the professional service work that is expected of me as a faculty member.)  It would certainly be possible for me to referee five times as many papers (although I would have to contact a lot of editors to volunteer my services), but that would turn refereeing into a the largest component portion of my total workload.  It would not be possible to do those reports properly and still be a productive researcher.