If overall legacy is being questioned, then I would argue that Konami is more deserving. Much of Konami's credit relates to arcade machines, not just home consoles, and they're with much more variety too. They're better known for long-lasting franchises whereas Square and Square Enix—besides a few prolific franchises—mostly focuses on more standalone titles or franchises with few entries, which is especially true in more recent years.
Below are notable franchises from both companies:
Konami: Beatmania, Bomberman, Castlevania, Contra, Dance Dance Revolution, Ganbare Goemon, Gradius, Metal Gear, Pop'n Music, Pro Evolution Soccer, Silent Hill, Suikoden, Tokimeki Memorial, Yu-Gi-Oh!
Square Enix: Dragon Quest, Final Fantasy, Front Mission, Itadaki Street, Kingdom Hearts, Mana, SaGa, Star Ocean