Elio's team composition says three specific things about him:

Elio has a subconsciously acute case of OCD via color coordination; he has a Pokemon of every color and type, and when each member is organized in a certain fashion, they parallel the colors of a rainbow. He has all his bases covered as a fashionista, and as both a veteran Top Coordinator and a multi-regional Champion-level battler.

He's capable of accepting Pokemon that have been shunned by society in one aspect or another and capable of bringing out their respective talents in lieu of their numerous differences and similarities - stating that even those who were rejected have something to offer and that no one is useless or a monster. Such openness and proper coordination not only exhibit his high degree of magnanimity, empathy, and creativity but highlights his status as both a genius and a charismatic intellectual - turning something that's originally perceived as a volatile bane into an awe-inspiring boon that drastically changes public opinion. This extends to individuals as well, capable of practically tolerating and working with basically anyone in lieu of temperaments, abilities, or origins - synchronizing with them to bring out the utmost results for whatever tasks lie ahead.

While not really team composition-related, Elio keeps all his Pokemon on him via modifying his Pokeballs. This shows that he not only values freedom - viewing the PC system as a type of prison, but he'd rather have everyone together: viewing them as an extended family while also exemplifying his inclusive nature. He wants all of his Pokemon to enjoy this long adventure with him, otherwise, it's not worth it.

(There's more symbolism to his team, but I'm too lazy to properly write it out. Sorry ¯_(ツ)_/¯ )