I suppose it's not impossible for another dragon to learn the shout he used, and it would work as long as the target dragon still had its soul stuck inside its skeleton, and not inside the stomach of another dragon.
The words of the shout, Slen Tiid Vo, translate to Flesh Time (Opposite of), and the Dragonborn learns Slen and Tiid as part of other shouts. I imagine it would be difficult for a mortal to understand the meaning of the "opposite of time" though, which I suppose might be like the untimes of the Dawn, as mortals can only perceive linear time properly, just like how dragons are unable to comprehend mortality.
The words themselves therefore might be implying just a simple undoing of time to restore the flesh, or it might be something akin to returning the body to the state it was in back in the Dawn, back to its prime when it was new and freshly shed from Akatosh. If it is more of the latter, I could see it being an ability reserved for Alduin himself as it seems like something only the largest, most powerful shards of Aka could accomplish, but the act of restoring flesh is more mundane, something a regular old dragon could reasonably do should he know the words of power. As for why they don't, it could be that Alduin was keeping the shout to himself, and would take down any follower of his who tried to use it, seeing it as an act of insubordination.