I've received a notification on my android phone and my chromebook apparently from Norton marked as "Norton 360 Service" asking for a sweeping range of permissions. I've never seen anything like this before and wonder if anyone else has had anything like it. When I tried to find the relevant page in the settings I couldn't find it. I can't find anything about "Norton 360 Service" online and when I tried to use Norton help it just took me round and round in circles!
My suggestion is that you received an update to your 360 app. That seemed to trigger a list of permissions that Norton 360 is requesting from your device. These permissions have always been there, but the update may have added or changed a permission, so the list was presented to you. For Norton I trust what it needs to access.
This same process will be used for other apps you install/update. So you need to look over the requested permissions and decide whether you trust the developer of that app to use those permissions safely.