If you have a Paramount+ subscription from subscribing directly with Paramount+, then you’ll have to watch their content in the separate Paramount+ app, and sign in there.
On Apple devices (iPhone, iPad, iPod Touch, Apple TV box), the Paramount+ app is able to “connect” to the Apple TV app, so that Paramount+ content can be listed in Up Next, Watch Now and Search in the Apple TV app, for seamless integration. The separate app is still needed, though.
On third party devices, others apps cannot “connect” to the Apple TV app, as that feature is supported through the system software on Apple devices.
The Paramount+ subscription as started outside the Apple TV app, is not valid for the Paramount+ Channel inside the Apple TV app. That is a different type of subscription with different terms and conditions, different perks and limitation, and not exchangeable as such.