Notification access is the only way Android lets one app see what another app is playing. The Play Store version of Carrio will only look at the YouTube and YouTube Music apps, and only at the title, the channel and how far into the video you are. That goes to your own PC when you tap, and nowhere else.
Android's warning talks about reading all your notifications, which sounds scarier than this. Carrio won't read them, and the app explains all this before it asks.
YouTube's app only shares the title and channel, not which video it is, so for this your PC has to look the video up, and now and then it shows you YouTube's results to pick from. Sharing never needs that, because the link already says which video it is.