A fresh postdoc able to secure an academic job joins as an "Assistant Professor". Fresh Assistant Professors usually cannot recruit PhD students, have limited funds to establish their lab, and no job security. It takes several grades and 15+ years to reach the position of "Professor", which comes with tenure.
I don't understand how you can go from being a postdoc to being a professor. That seems to really diminish the value of the title 'professor' to me, if people can get it as basically their first permanent academic job.
In my home country (India) the academic job hierarchy used to be "Lecturer" --> "Reader" --> "Professor"
Somewhere in the last decade or so this nomenclature was revised to "Assistant Professor", "Associate Professor", and "Professor". I can't speak to the rationale for this, but it brings it in line with the naming convention in the US.
While the minimum requirement for an Assistant Professor job in India is a Masters, in practice now you really need at least 2+ years of Postdoc experience. To move up to an Associate Prof. position, the minimum requirement is at least 5+ years of post PhD experience, to have guided at least one PhD student, and n number of publications with x Impact Factor. A Professor requires at least 10 years of post PhD experience, and at least 3 PhD students. Of course, these are minimum requirements, and I know of several people who far exceed those qualifications and still waiting to move up the ladder. For example, a senior from my graduate lab has been an Assistant Prof for about 14 years now, has 15+ publications, and 3 PhD students who've gained their degrees.
For what it's worth, the job titles can be ambiguous.
The formal progression in the US/Canada is usually "Assistant Professor" -> "Associate Professor" -> "Professor." However, someone in any of these positions can be called a (lowercase-p) "professor", with the most senior job referred to as a "Full (or University) Professor."
They can, just not right away. An Assistant Prof. in Philadelphia I had interviewed with for a postdoc position told me that since his lab is new, the school won't let him recruit PhD students for the first year. He can take postdocs and trainees, though.
I think there's also a practical problem in attracting students to a new lab. Another Assistant Prof I'd interviewed with at a premier research institute in New York had established her lab over 3 years ago, and still hadn't managed to recruit either a PhD student or any postdocs. She may be an outlier, of course.
I really wish Tulane had enforced that on the two starting assistant profs I tried to work for before they took their NSF CAREER awards and ran as far away from New Orleans as their legs could take them. You were in biomedical research or math?
