The short answer is: Yes. It happens to some people. Not to all, but to some. I am one of them, and I now know a few others who experience this with regularity - always have. The evening after a strenuous ride, I will often run a fever and might even develop a sore throat. While it's happening it can feel very much like the flu. But the next morning all symptoms will be gone, so it is not a case of actually getting sick.
My understanding is that this is a normal reaction that some people simply have to certain types of exertion. It may have to do with how our bodies engage in muscle repair. Or it may have to do with circulation issues. No one seems certain, but it does happen.
Regardless of how or why it happens, I have noticed there are things I can do to alleviate it. For instance: taking a hot shower or bath after a ride, taking an NSAID or acetaminophen concoction, eating warm "comfort food" such as soup or scrambled eggs, drinking lots of fluids, and trying to get as much physical rest as possible. Basically treating it like the flu works for me. It dulls the symptoms while they are happening, so that I can still be productive with the rest of my evening. And the next day I feel good as new - only the muscle soreness remains.
If you experience fever or flu-like symptoms after strenuous cycling, how do you deal with it?