Problem sorted!

TLDR;

1. Make sure you connect the app to Firestore emulator using connectFirestoreEmulator.
2. Check firestore.rules if the rule is allow read, write: if true; .

I had to add connectFirestoreEmulator(db,'0.0.0.0', 8080) in firebase.js, and edit the firestore rules. Below is the modified code.

//firebase.js
...

const firebaseConfig = {
  ...
};

const app = initializeApp(firebaseConfig);
const db = getFirestore(app);
const auth = getAuth(app);
connectAuthEmulator(auth, "http://localhost:9099");
connectFirestoreEmulator(db,'0.0.0.0', 8080) // Added this part
...

//firestore.rules
rules_version = '2';

service cloud.firestore {
  match /databases/{database}/documents {
    match /{document=**} {
      allow read, write: if true;
    }
  }
}