That's literally not how that term works though, a third party is a character that appears on multiple platforms not just one is a character in associations with more than two companies. Hence the name THIRD party. Geno is owned by a studio that's a third party developer sure but that doesn't mean everything they make is third party. With that logic you're basically saying Cloud was never a first party character back when FF7 was a first party Sony game. Geno is and always has been second party not third party. A second party character is a character that appears in an EXCLUSIVE console game/games but who's character's in said game/games are not owned by the console parent company which is literally what Geno is. He is from an exclusive Nintendo IP that is not owned by Nintendo and a character who has never appeared outside of said exclusive game since after.
Whether something is first or third party has nothing to do with whether something is exclusive to any given console. The second party is the consumer. There is no such thing as a second party character.
Nintendo owns Mario, and sells you a game with Mario. Nintendo is the First Party, you, the consumer, is the Second Party.
Nintendo does not own Geno, and sells you a game with Geno. Nintendo is the First Party, you, the consumer, is the Second Party, and Geno's rights holder, Square, who is not involved directly with the transaction, is the Third Party to that transaction.