To a certain extent, yes. They have a brand, and they attempt to protect it. Once you qualify and participate in the tour, you agree to their rules regarding appearances in unaffiliated tournaments.
This isn't limited to professionals on the tour either. The PGA is also basically a guild that sets the standards for professional course/club managers and teaching professionals, and keeping a job as either while maintaining PGA affiliation means abiding by their requirements as well. Choosing not to maintain PGA affiliation generally means you will not be hired by reputable golf courses.