You may not have another app open, but you may have a service running that does this. Either one that starts with the machine, or one that is left over from a program that was previously running.
You can eliminate the latter by rebooting, and seeing if you can get Now running from a clean start, but it won’t fix the former.
You could usefully search the Community for similar reports; some software called Panopto has found to be the culprit in several cases, and other software is mentioned; see if you recognise any of the names mentioned.
A useful tool here is Activity Monitor.
I’ve sorted mine by Process Name, but there are 440 to go through, so the Search, top right, where you can put in things like ‘screen’, or ‘capture’, may be of help.
One of my colleagues on the Community, @redchiz1 , is much more of a Mac expert than me, so I have included him to see if he can help here any further.
@RoyB Flatters me somewhat, but sadly my Mac knowledge does not extend to the use of the NOW player, my MacBook is not my go to choice for viewing.
This has come up before from vague memory though, if there is anything that NOW detects as screen sniping/recording software it will not play nice, do you have anything installed that may fit that bill? Have a search across the forums and maybe elsewhere.
@Alexander Could be something simple and mentioned before, or unfortunately something not yet mentioned and unique to you. Have a look through the linked thread for some suggestions.