According to Zordon his Green and White Ranger powers were lost "forever" when each left, respectively. The Zeonizers / Zeo powers were never destroyed, however. They were just discarded in favor of the Turbo powers. So Tommy's probably held on to his Zeonizer since then. The power from the black Dino Gem was used up by the last episode of Dino Thunder.
All this being said, it was established in the two parter "Once a Ranger," from Operation Overdrive, that powers can be physically restored in the morphing grid. Tommy's spent a fortune on things like the Astro-Megaship II, dinosaur research...etc...etc... So it's not unfathomable that during this time he found the morphing grid and restored some of his old powers.
The Ranger keys in Super Megaforce also seem to channel all of the powers from the first 19 seasons. So it's possible the keys have something to do with re-energizing his powers.
The interesting thing about this clip is that even though Tommy seems to be channeling the White Ranger powers, it's confirmed he's actually going to morph into the Green Ranger later on:

So maybe he just restored the Green Ranger powers and kept Saba around out of sentiment. After all, the White Ranger suit and Saba were on display in the Power Chamber during Zeo. Maybe he just took it home. Just some guesses.
