iTunes is a protected brand no one else can use. It is pretty much impossible to register something so generic as "Online Media Store". Apple even failed to protect the word "AppStore".Rebrand iTunes. Its ridiculous to go to an online media store to buy and rent movies and videos and have it called iTunes. So, how about re-branding iTunes as a radio/music only product and then come out with a new brand for Apple's walled media garden.
Apple's false advertising claim against the Amazon Appstore thrown out by judge
So when Apple rebrands the iTunes Store to Apple MediaStore, nothing will stop the competition from opening their own Amazon Mediastore. Unless you can come up with a very specific name for a store selling unspecific media products, it is better to stay with iTunes. Accurate naming and legal brand protectability don't go together.