I would love to know the circumstances under which people opt for a second doctorate degree.
Could a professor working in a university write up a thesis and submit it to his department to earn a second doctorate?
Do only those who want to remain in the university forever as students go for a second doctorate?
Are there any general merits/downsides to earning 2 PhDs, if you are perhaps assured of a research career after a good first one?
Generally you cannot get two PhDs in the same field. One could get a second PhD in a second field.
People who want to do research in multiple fields or in a multidisciplinary topic or wish to change fields may obtain multiple PhDs. Alternatively, as you suggest, people who want to remain students forever do that.
The main downside of doing multiple PhDs is that people may not see that you are moving on with your career. There is a career after getting your PhD, and obtaining a PhD is a small step along that career path. Hovering around getting multiple PhDs would be akin to getting multiple bachelor degrees. People may see you as a permanent student, not someone growing into a mature researcher.
I think it really depends on what the person actually wants to do with the two different degrees. For example if someone has both a PhD in chemisty and biology to become a biochemist, their understanding of the subject would be much more versatile. It is absolutely not nessesary by any means, but it would just depend on the person and how many angles they want to approach a subject; some people prefer more than one way of looking at something.
Most universities in the US prohibit awarding of duplicate or comparable degrees. For instance; one might have earned a PhD in Management from Papua New Guinea University, and planning to earn another PhD in Management from Yale. In this particular case, Yale won't admit that student. Similarly, HBS, NYU Stern, Berkeley Haas MBA program FAQ sections clearly state that they won't accept students who already hold an MBA degree from another institution.
"University policy prohibits awarding of duplicate degrees. If you have an MBA or comparable degree from an institute of higher education, your application will be ineligible for consideration."
Most schools don't like duplicate degrees, especially ones in the United State. Schools in Africa or the UK usually allow it.
I know a Professor that had 5 doctorates (not honorary doctorates).
Professor Blight (PrEng) was an A-Rated researcher who held five doctoral degrees, and who was considered as one of the world’s leading thinkers in Geotechnical Engineering. He held BSc(Eng) and MSc(Eng) qualifications from Wits. He completed his PhD in Soil Mechanics and his DSc(Eng) in Geotechnical Engineering at London University, before returning to Wits. In 1985, he obtained his DSc(Eng) in Materials Engineering from Wits and another DSc(Eng) qualification from the University of Cape Town. In 2001, he obtained a D.Eng, degree from Wits focusing on the Application of Research in Practice.
With over 330 papers published in accredited journals, Professor Blight was internationally renowned as a researcher of the highest standing around the world. He was well published internationally and the author or co-author of several books.
In answer to this part of the question:
"Could a professor working in a university write up a thesis and submit it to his department to earn a second doctorate?"
Where that is allowed he would need to have been registered as a PhD student for the required period of time, which in many British institutions is three years if the research is full-time and longer if it isn't. Universities can't just dole out PhDs at the drop of a hat. The department and probably the higher degrees committee would also want to know in what framework he had carried out the research. I think he would have to do it outside of what he gets paid to do as a professor.
I am not sure how easy it would be to get round the requirement that a PhD student should have an appointed supervisor during those three years either. Of course he may not wish to do that, but if he did there might possibly be a way at some institutions.
