You started the correct game. The characters you're talking about are ex-DLC mixed into the original game, and due to the self-awareness of the series there's a few references to the fact that it's a remake, and a reference to the second game you won't understand (it's for the best). There are a few jokes slipped in for longtime players.
Dunno the deal with Falcom, possibly because she was DLC in the first game?
Neptune calls Brocolli Puchiko because...Brocolli is literally
Puchiko from one of Brocolli's IPs.
I forget with MAGES, her whole deal is she's time and space blah blah blah, I think it's a Steinsgate reference?
As for Falcom representing a company, there are two kinds of characters:
CPU (Canidates) and Makers. The CPUs represent game consoles (or console brands at this point), Candidates represent the portable brands, and Makers represent game making companies. Generally the Makers will be named directly after a company and they're licensed directly by that company (not 100% sure how this all works...but it seems like NIS, not CH, actually owns NISA for instance?). Outside of the Oracles in game 2 (and Histoire in all games) every playable character is a company reference, and most villains are references to piracy, specific pirate hardware etc.
The Makers are often DLC but most DLC characters were integrated into the story for the rebirth games--in fact almost all prior DLC is included via ingame unlocks now.
