I would say I am strongest in southwest/Latin cuisine and Japanese cuisine. I was classically trained on the job in French cuisine which is a great foundation.
Yes a a private chef clients do like a certain style of food. Just look at your family do they eat a certain style of food? You would have to be able to cook for the client what they ask for. A very up scale dinner party for them to mac & cheese for the kids. As you learn to cook you will have your on style from meat and potatoes to vegetarian. As you develop your style the clients and you would become a match. I did not stay a personal for very long. I found that it did not work for me. Right now I enjoy cooking a wide verity of foods. My job lets me do just that so one day I get to do Greek food next ramen, another day just a good old fashion burger. That's what I enjoy. Do not rush into your cooking choices learn and try many things. it takes time and does not happen over night.