I think it depends on why someone is a vegetarian. Strictly speaking, of course, a vegetarian does not eat fish. But there are many people who have cut red meat from their diets for health reasons and then it's an easy next step to chicken but fish is seen as more healthy than either of these two.
I am not currently a vegetarian but I was for several years. One of the reasons that I returned to eating meat was that I needed some balance in my life. I found that once I started cutting out meat (all of it, including fish) on principle, eggs came next, then all milk products, then of course I couldn't wear leather, and pretty soon I wasn't eating certain vegetables and no wheat or yeast. It got to be pretty difficult to feed and clothe myself.
Peace.
~ ~ ~ All meetings end in separation All acquisition ends in dispersion All life ends in death - The Buddha