Do researchers study topics outside their field for fun?

Fundamentally, a researcher studying a different field "for fun" is in no other situation than anybody else studying "for fun". It's a hobby, just like reading, going to the theatre, or sports. Some engage in this hobby, some do not. Whether they "have time for that" is really the same question as whether they have time for any other hobbies - some do, some (especially on the tenure track) are maybe a bit crunched for time, and choose to use their little spare time on something a bit farther removed from their daily work grind.

Is there any known case of a researcher who, by studying a topic outside his/her field for fun, had an insight for a novel approach or solution?

I am sure there is. Note that "having a an insight" is not a particularly high hoop to jump over. My students "have insights" all the time, the question is whether they are important, work well enough, and are generally more useful to pursue than the myriad other ideas they could use.

My gut feeling is that you rather meant whether somebody had a big breakthrough in a different field as a "hobbyist". I think this used to be more common in times past. I can't really think of a recent example (at least in the STEM fields), as plenty of time and resources (both of which a hobbyist definitionally does not have much of) is required for major breakthroughs.

Time is a serious constraint for researchers who want to keep themselves competitive. You need to write grants, advise students, go to pointless but mandatory meetings and you may need to teach.

Nonetheless, a lot of people I know have hobbies outside work: cycling, singing, language learning, etc.

Research as a hobby for a researcher is a tough proposition. As a professional researcher you tend to hold your hobby research to the same standards as your paid research, and that takes time and dedication. But, I've seen people doing this. There are physicists doing hobby research in history, economics, or mathematics, or other branches of physics completely orthogonal to the research they are paid for. My own student is doing well enough with his PhD work, so he decided to take on field theory in his spare time.

Not everyone is able to set aside time for pet research projects. I, myself cannot. If I work on a project, I usually devote all my research time to it. The only way for me to do hobby research is to steer my research towards subjects I'd like to work on. In other words, do as much hobby research as possible at work.

It depends what you mean by "study for fun".

I would like to think that many researchers will find what they do professionally as "fun", and so do not feel the urgent need to do something not in their field of specialization.

The most famous of all examples are the contributions to physics from a patent clerk named Albert Einstein; he did have formal training in physics but was not a faculty at that time, and so presumably was still studying physics "for fun".

One of the early breakthrough results of quantum information theory is the famous BB85 protocol for quantum cryptography, by Charles Bennett and Gilles Brassard; neither are physicists by training, although Bennett worked at IBM at the time (and still does) and Brassard specialized (and still does) in cryptography. Both knew physics, and I don't think they brushed up on their physics entirely "for fun", but this example would fit the kind of answer you are looking for.

Depending on your you define "field", you will find many researchers who are accomplished musicians and have studied music extensively, for instance. Although not professionals, some of them are pretty darn good.

In the case of languages, the real barrier is lack of occasion to use "for fun" a language you would have learned. There are plenty of "teach-yourself-German" books, but if you cannot interact in that language in your environment it is unlikely to be much fun.