Within several European countries they do, since the banking systems support sending a payment using a phone number. You get confirmation of the recipient's name before pressing "send".
It's MobilePay in Denmark and Finland, Vipps in Norway, Swish in Sweden, PayM in Britain, etc.
For example, we split a restaurant bill at the weekend. In roughly the time it took one person to pay the whole bill on a card, the rest of us had sent ¼ each to her using her phone number. There are no fees for this.
In the next year or two, this should start to work internationally.
https://en.wikipedia.org/wiki/European_Mobile_Payment_System...