I was in a very similar situation. When I was about 25, I found out that I was born with only one functional kidney. (There is actually a second kidney, barely visible under various imaging techniques, that looks like a dried-up string bean, so the diagnosis is generally "atrophied kidney" rather than "solitary kidney," but I don't believe that the difference is particularly important. I wouldn't be surprised if you also have a second, but atrophied and nonfunctional, kidney.) I suddenly felt very vulnerable; I even remember being extra careful that day crossing streets while walking home, because I suddenly realized that I couldn't afford to let my one good kidney get banged up in an accident.
The discovery that I had just one functional kidney was accidental; a physician just happened to notice it while reviewing a bone scan for a possible metatarsal stress fracture (which I did have, by the way). I suspect that most people learn about these conditions by accident, since they are likely to be asymptomatic.
At the time, I was already an experienced runner. I had run cross-country and track in high school and college and had run in many road races after college, including several marathons, with a best (at the time) of about 2:23. Like you, I was passionate about running, and returned to it after my metatarsal fracture healed. In later years, I trained harder and ran faster, with a PR of 2:18 in the marathon. After quite a number of years away from the sport, I returned to competitive running in my forties for a short while, running a 2:28 marathon when I was 41. If there were any problems caused by my lack of a second functional kidney, I wasn't aware of them.
In my fifties, a number of odd circumstances rather suddenly manifested themselves -- including a couple of low-impact bone fractures, very low bone density, bladder cancer, an accumulation of kidney stones, and even gout (accompanied by high uric acid levels in the blood). Especially worrisome to me was a very rapid rise in serum creatinine, which generally suggests a rapid deterioration in kidney function. As you may know, kidneys play an enormous role in the proper functioning of the body, including bone metabolism, so the relationship between an aging, solitary kidney and a wide range of problems should not be ignored. I have, however, had the good fortune to get the attention of some outstanding physicians, and am quite optimistic about my future health. Even my serum creatinine levels have come down to something near normal range, which is pretty good work for a solitary kidney.
My bottom line for you is this: It's not the end of the world -- not even close. I doubt that your solitary kidney will interfere with your running, and I doubt that your running will cause major problems for your kidney. You should be aware of the possibility that your solitary kidney could predispose you to certain problems over the very long term, but that can be monitored. Oddly enough, the single most important thing you can do for yourself is probably the simplest: Drink more water. That's probably a good idea for most people anyway.