How do I get an object from local storage after json.stringify()

I am currently creating a login page with ReactJS. This page utilises an API which has an 'authorise' function confirming whether the entered details are authorised to use the API. To handle the API calls, I have created a class which has a function for each possible API call. If the user is able to login successfully (if the authorise function returns a status of 200) they are redirected to a 'ProjectSelect' page/component. From this Project Select page the user will be able to select a specific project from their project list. Now, here is where the problem lies. I need to now use the API instance from the 'Login' component in the 'ProjectSelect' component. The way in which I am currently trying to do this involves storing it in local storage as such:

Login.jsx:

import {API} from 'filepath';
const api = new API();

async function loginUser(email, password, appid) {
  return api.authorise(email, password, appid);
}

export default function Login() {
  const classes = useStyles();
  const [email, setEmail] = useState();
  const [password, setPassword] = useState();
  const [appid, setAppId] = useState();

  const handleSubmit = async e => {
    e.preventDefault();
    const response = await loginUser(
      email,
      password,
      appid
    );
    console.log(response);
    if (response.status === 200) {
      swal("Success", "Logged In", "success", {
        buttons: false,
        timer: 2000,
      })
      .then((value) => {
        localStorage.setItem('access_token', response.data.access_token);
        localStorage.setItem('api', JSON.stringify(api));

        window.location.href = "/bimpluslogin";
      });
    } else {
      swal("Failed", "Login Failed", "error");
    }
  }

ProjectSelect.jsx:

const api = JSON.parse(localStorage.getItem('api'));
let allProjects = api.getAllProjects(); // Error occurs here 

I am aware that if you JSON.stringify() an object it will only take the key values leaving it basically unusable when parsed back through the JSON.

What I've tried: I have tried to create a new API instance in the ProjectSelect component and storing the auth details in local storage however that causes a lot of issues and doesn't allow me to use all of the functions. I would appreciate any suggestions on how to store an object from one component and then use that exact same object in another component. Thanks in advance.