Craft CMS Backing React FE using the element API
I created a react native app with Craft CMS (still version 2.6) as backend using the Element API. But you're right, user authentication is not supported by the Element API. However, I created my own plugin to get user authenticated. I even create user and update their details through the app. 

But storing user sessions in React Native is another story. That has nothing to do with Craft or any other CMS system cause you can't get server session to React Native. You need to write your own session handler. I remember a good blog post about that topic: https://auth0.com/blog/adding-authentication-to-react-native-using-jwt/. 

There also might be other solutions because this could be a little bit overkill.

Hope this helps you as a first start.