It affects how you ride the bike, yes. For a simple example, if you ride with a "lean bike" riding style, you'll be able to switch directions very quickly and easily, which is great for chicanes. On the other hand, if you ride with a "lean body" riding style, you'll be able to carry more speed through a corner than someone with a "lean bike" style.
In real life, racing riders use both of those styles over the course of a lap, depending on the corner. That's why some have criticized the fact that we're stuck choosing one style per race.