Each physical device is purchased (obviously). If each is set up on the same Roku account, then they both will have the same channels available. Some channels may require you to log in on each device and some will propagate authorization to the new device automatically. (Assuming you set them up on the same Roku account.)
Some channels have their own limits such as a maximum number of streams that can be playing at once. Generally, you would expect any such limits or authorization issues to come with an error message, not just a blank screen.
Do all channels produce a blank screen or just Paramount+? If all, then I would be inclined to think that you have something defective or incompatible – possibly the Roku, but possibly also your HDMI port in the TV, or the TV might be incompatible in some way – the only way I know to debug stuff like that is to move stuff around – ie: try different ports on the TV or another TV. If the Roku produces blank screens on various apps, connected to various TVs, then I would swap it for another one at the store.
