My answer to this question is yes and no.
No, because as a professional you won't be asked for proof of an arts degree, all that matters is your portfolio. An art director hires you based on the work they see in your portfolio, not your education. If you have a lot of drive and self discipline it is possible to teach yourself art from books and online courses. However, if you need more guidance and aren't disciplined enough to learn on your own, I would recommend art school. Art school also gives you time to experiment with different media and to develop your own style. I went to art school because I had an idea of where I wanted to go, but wasn't sure of the details of how to get there. School provided structured learning but also freedom to choose the courses I wanted. Most art schools have a foundation year, where you should be taught the fundamentals of drawing - anatomy, perspective, etc. From there you can focus on what interests you; for me that was printmaking and illustration. Instructors can also guide you in deciding which area of illustration to go into. I took several different illustration courses - such as decorative, concept art, comics, editorial - to help me explore the different areas of illustration. I was encouraged by instructors to aim towards editorial, based on the work I was doing. I also learned that concept art was not for me; I learned that wasn't where my strengths were.
The goal to aim for as an illustrator is to develop a personal style that is consistent. This is key because art directors need to know what to expect when they hire you. If you have inconsistent work, in a variety of styles, you appear as a mixed bag, and art directors won't know what they will get when hiring you. In editorial, art directors match style to articles, so style is very important.