Where do we store user data after login in React Native

I suggest separate the authentication block from the rest of your code. Keep it simple by using a Context and Provider component:

import React, { createContext, useEffect, useState, useContext } from "react";
import AsyncStorage from '@react-native-async-storage/async-storage';

import { logIn, logOut } from "../services/api/auth";
import { loginAndSaveUser } from "../utils/Auth/loginAndSaveUser";

const AuthContext = createContext();

const AuthProvider = ({ children }) => {
// this provides current session data to its children
const [user, setUser] = useState(null);

// auto login when opening the app
useEffect(() => {
    _autoLogin();
}, []);

const _autoLogin = async () => {
    try {
        // this read data from storage
        const user = await AsyncStorage.getItem('user');
        if (user) {
            const response = await logIn(user.username, user.password);
            if (response.ok) {
                // now you provide the user data to children component
                setUser({
                    username: String(username),
                    password: String(password),
                    userId: String(response.text.userData.id),
                    token: String(response.text.accessToken)
                });
            } else {
                // this means the data in storage is not right
                await AsyncStorage.removeItem('user');
            }
        }
    } catch (error) {
        console.log(error);
    }
}

const _signIn = async (username, password) => {
    try {
        const response = await loginAndSaveUser(username, password);
        if (response.status === 'success') {
            setUser(response.userData);
        } else {
            setUser(null);
        }
        return response;
    } catch (error) {
        console.log(error);
    }
}

// choose logout -> delete data
const _logOut = async () => {
    try {
        const token = user.token;
        if (token) {
            const response = await logOut(token);
            await AsyncStorage.removeItem('user');
            setUser(null);
        } else {
            setUser(null);
            console.log("Error in logOut function (Auth): No token found for current user");
        }
    } catch (error) {
        console.log("Error in logOut function (Auth): ", error);
    }
};

const value = {
    user,
    _signIn,
    _logOut
};
return (
    <AuthContext.Provider value={value}>
        {children}
    </AuthContext.Provider>
)};
const useAuth = () => {
    const context = useContext(AuthContext);

    if (!context) {
        throw new Error('useAuth must be used within an AuthProvider');
    }

return context;
};

export { AuthContext, AuthProvider, useAuth }; 

and in children components, for example the login screen:

const auth = useAuth();

and use the data:

const handleSignIn = async () => {
try {
  if (!email || !password) {
    Alert.alert('Missing Infor','You may miss some fields');
    return;
  } else {
    const res = await auth._signIn(email, password);
    switch (res.status) {
      case 'success':
        break;
      case 'failed':
        ......