The phrase "exclusive beta" refers to a limited trial version of a product or software that is only available to a select group of people. Full definition
The game will have a 30 day exclusive beta on PS4. (stevivor.com)
In addition, a 30 day exclusive beta only on PS4 will be implemented. (apgnation.com)
In addition, receive exclusive Beta builds of the game to see how the game is developing and help the development team shape the game through your feedback. (kickstarter.com)