We've already gone through a number of solutions, and regardless of the tech you built, this is just another example of an app-approval flow. Twitter & Facebook allow you to approve logins from already approved apps, and while their underlying tech is different from yours — your server can still change the tech maliciously.
For a brilliant app idea, see Kryptonite[1]. It lets you install a browser add-on that pretends to be a U2F key, but the private keys are stored in your phone's secure key storage (Keystore for Android & Secure Enclave for iOS). The experience is similar to what I assume your approval from a phone would be, but it works on all website that support U2F.
For your own site, I'd recommend using U2F tokens and TOTP + OTP keys, they're considered really good solutions, and while SMS is considered bad — it's still better than no 2FA. Please see the post by Troy Hunt[0] on 2FA from late last year. It covers more than I possibly could.
Sorry for the negative critique, but I'm sure you'll build something great next time ;) Instagram was first launched as a Foursquare clone[3]
[0]: https://www.troyhunt.com/beyond-passwords-2fa-u2f-and-google...
[1]: https://krypt.co/
[3]: https://www.theatlantic.com/technology/archive/2014/07/insta...