The truest answer is, “it is personal”. It is not something that has to be done nor required for regular advancement in an association or club. It does not necessarily change what you know, but may, or should change how you perform. One thread runs through all martial arts, that the goal is towards constant improvement. Perfection cannot be attained but is pursued non the less. With that in mind my performance from one association exam to another should be better with each attempt, subject to the amount of time between them. The question remains, Why examine in multiple organizations?
For the challenge.
In most martial art associations, there is a minimum amount of time at your current rank required before being eligible to examine for the next level. The higher the level, the greater the time between, usually in years. I see many who basically cram for the exam, especially when there is two or more years between. Regular practice and workouts are allowed to slip by the wayside as no immediate challenge is there to warrant keeping up the training. (Also proves as an argument in favor of competition but that’s a different topic) If the various association exams are spaced apart, you could always be training for the next goal for many years to come.
Using it as accreditation.
You have seen or heard others speak of it. Intimation that an achievement was bought or “given” to a person. That “That Association” is known to give ranks easy, or for a price. For me, this is where the value of multiple association exams come into play. If my association awards me “X” degree, there may be those who question the circumstances of the award. Am I the favorite? Am I in their pocket? By choosing to examine in other groups such as Karate Canada and Japan Karate Federation Goju Kai it becomes clear they agree with the previous award given and puts any other considerations in their place.
So whatever reason you choose to or not to, it really is personal.