IIRC, That simply awakened the dormant demon blood within him. Similar to how Dante did when Rebellion activated. Only difference is that since Dante's Sparda's son, he's naturally far stronger than Nero.
If you go back through all the games, aside from the Mundus battle, and his awakening in DMC3, Dante practically never actually Devil Triggers within canon. (which is why people bitched about the anime)
The closest thing to it that ever happened in 4 was a slight resonance with Nero when their arms touched. But that was it. Even on Dante must Die mode against the AI Dante, he used all his weapons, styles, and move sets but didn't DT.
If I'm not wrong Spiderman is also a background character again (that super-jumping photographer in the NYC stage HAS to be Peter Parker) and you don't see him getting left out of the game.
There's 2 doing that. Both of them wearing suits which Parker doesn't do, and also one of them is short and fat.