It's a PlayStation exclusive game series, but the Omega Collection is not in first-party development
since the WipEout developer was closed in 2012. The Studio developing the Omega Collection is a
third-party studio.
This also applies to Crash Bandicoot. Its a PlayStation exclusive games series formerly developed by
Naughty Dog. But the developer of the Crash N-sane Trilogy is Vicarious Visions and part of Activision.
This means the N-Sane Trilogy is in third-party development too.
If the developer is not part of Sony in any way its exclusive but not a first-party game regardless of the IP.
First-Party means the publisher is the hardware maker. Third party means licensee. That's the cleanest definition.
The developer being owned or partially owned or not-owned doesn't mean anything. SONY could own or partially own a developer, yet a licensee could publish the game.