This post on another thread is generally along the lines of what I've seen offered before as an explanation whenever this question gets asked:

Quote Originally Posted by Rongway View Post
The cooldowns persist so that you while you're overworld questing you can't burn all your powerful instant attacks, change jobs away and back, and then burn all your powerful instant attacks again. So the cooldowns need to be kept track of even after you switch jobs.

But if the cooldowns weren't shared, then you could ostensibly burn all your instant attacks on RDM, then switch to DNC and burn all your cooldowns, then switch to NIN, then to .... etc.

It wouldn't happen very often, if at all, but the SE devs have always been at the extreme end of software design conservatism. They plan for the highest possible volume of data access and transfer, and the highest possible computational commitment, and take steps to minimize both, even when such extremes would be unlikely. Ergo limiting the number of cooldowns each player can have going at once by sharing them between jobs.