Kingdom Hearts III Plot Twist & Future Setting

I think he trained his heart just like Sora did at KH 1, his travel made his heart strong enough so that he could take the keyblade back from Riku.

Axel had been exposed to both Roxas' and Sora's Keyblade (by that time his "new heart was awakening") . Being a Nobody is a living proof that your heart was strong (had a strong will), when he became human (Lea) he was half way done in his path to be a keyblade wielder.