Maybe this chart is a little misleading but I looks like Mastercard and Visa are quite superfluous.
Why do they exist?
Visa and Mastercard are cards owned by the banks. They take a cut mostly because they can, but also because the infrastructure involved to process a transaction isn't free. For what it's worth, non-bank cards like American Express take a larger fee.
I don't understand what you mean by "Visa and Mastercard are cards owned by the banks."
You can't go to Visa/MC and get a card. You have go go through an issuing bank. As opposed to Amex where your card is issued from them directly (though, they do have bank issued Amex cards too).