Noppe
First party studios and games are made by Sony owned studios.
Hence first party.
Second party studios are not owned by Sony but are making only games for Sony cous they have a contract in some form.
But that Contract can end ore bought off, just like at that time the second party studio Insomniac. They where a few years an indepented Studio and made a game for MS under contract for one game.
Now Insomniac is a first party studio owned by Sony.
Games are 1st / 3rd party. Not studios. And 2nd party games don't exist.
The 1st party legally involved when you buy a game is the console manufacturer / platform holder (like Sony).
The 2nd party legally involved is the user buying the game (User).
The 3rd party (that can be) legally involved is another publisher (like Activision).
If a game is published onto the console platform by the platform holder (Sony), then its "1st party." Only Sony and the user are involved in the deal. No 3rd parties are involved.
If a game is published onto the console platform by anyone other than Sony, then a 3rd party is now involved in the legal transaction of you buying the game. So the game is "3rd party." Because all 3 parties are involved legally in the deal when you buy a game. Sony, the user, and Activision, in this example.