In one of the gameplay footages at E for All, Fox was inside his Landmaster Tank when another Smash Ball appeared. The tank simply touched the Smash Ball, and the ball broke immediately (due to the large amount of damage the tank does when it runs over something). After Fox exited the tank, he was glowing again (the same glow you have after getting the ball but before using the Final Smash). This comes to show that a Smash Ball can appear while you do a Final Smash (and maybe even before it if you hold onto it for long enough without using it), and also shows that one cannot use a Final Smash while currently doing their Final Smash. But through the first case, it may be possible for Final Smashes to clash. In the case of Super Sonic (SS) vs. Giga Bowser (GB), SS would simply be able to hit and damage GB, but GB will not flinch. And since SS is invincible (even from damage), GB can't really do anything except to TRY to avoid SS's attacks (which can prove to be quite difficult, since he's so huge).
PS: It's too bad you can't do a Final Smash while currently doing one, or else they can have an Easter Egg where if Super Sonic uses a Final Smash, he'll turn into Hyper Sonic (probably with no change in attributes except for visual effects).