Title.
MP is a resource that every combat job has access to. At FFXIV, MP is used to represent the Aether you use to cast some spell. Now, MP management is a mechanic that's used only by magic users, notoriously Disciples of Magic (aka Magic Ranged DPS and Healers), but Paladin and Dark Knight uses it.
But the story let us understand that physical skills uses aether to be made. For instance, Gunbreakers cartridges is filled with aether. Machinists uses Aether to power up their devices. Ninjas uses aether to do their Mudras. Dragoons uses the dragon's aether to empower their jumps. Monks uses aether to open their chakras.
MP is the perfect resource to represent the use of Aether to augment physical capabilities.
As for abilities, they're just spells and skills that's not tied to GCD. They do same thing as a spell/skill does. So there is no reason for them to be free. And it opens up for some mechanics that hurts MP instead of HP.