What would be the possible pros and cons of having Inspirations owned by Characters instead of players?

Well, you limit the affect a character can have on the story (mechanically) to just whenever it is in a scene.  If Inspirations are owned by players then an Inspiration like "Galaxy Girl proves that might makes right." can have an affect even when Galaxy Girl isn't in a scene.  Thus, the themes and actions that occur during play have continuity.  If you make it so Inspirations are tied to characters then those same themes and actions are only relevant when that character is being played.

I'd suggest, strange as it seems, that such a rule goes along well with imagining that the characters have theme music.

When Galaxy Girl enters the scene, the whole world becomes a little bit more the way she sees it.  Her past acts and successes have an impact.  Her theme music is playing in the background.