Street gangs have the same goals and core beliefs. Each gang wants to control its neighborhood and has rules governing how this will be accomplished and how its members will act. Since rival gangs may be competing for control of the same neighborhood, gang loyalty is paramount and one cannot belong to two different gangs.
All Taekwondo organizations, the WTF and ITF for example, teach Taekwondo and have the same core beliefs, but they each have variations and differences and as to the way Taekwondo should be taught and performed, and each wants to control Taekwondo around the world. Some of these variations and differences are not compatible with each other. For example, if you teach that techniques in patterns must use low stances, use obvious chambering, use hip snap with punches and kicks, and left the rear heel at impact, and you really believe this is the correct way to achieve power and perfection of technique, then you cannot reasonably also teach that using high stances, no chambering, leaning to punches and kicks, and not lifting the rear heel lift will also generate power and be technically perfect. Just as with street gangs, instructors and schools cannot profess loyalty to two rival Taekwondo organizations and survive within in either organization.
When you believe in two opposing beliefs, you believe in nothing. If you really believe something, you may be considerate of a rival belief, but you cannot accept the rival belief. If you are a Christian and believe the only way to heaven is through Christ, you may be considerate of Muslims, but you must also believe Muslims will not get to heaven unless they accept Christ as their savior. An instructor may have been a black belt with the ITF and then changed to the WTF and earned a black belt with that organization, but the only way to be a true WTF black belt is to reject the ITF and accept the WTF as the true leader of Taekwondo and to only teach Taekwondo as dictated by the WTF. If an instructor tries to believe in both organizations, he or she truly believes in neither.