Very wrong implication.
A publisher can publish a game without owning the IP, and if that publisher is a platform holder, it means the game is still 3rd party. IP ownership determines if a game is 1st party or not.
Example: PlayStation was the publisher for Call of Duty in Japan for many entries. It was still a 3rd party game.
Another example: Microsoft published Mass Effect in the past. It was still a (timed exclusive) 3rd party game.
Another Example: Sumo developed Sackboy Adventures, an IP which belongs to Sony. It was a 1st party game.