Firebase - check if user created with Google Account is signing up or logging in?

There aren't any separate methods to login and sign up when using Google Sign-In or any other provider. If a user with that credential exists then user will be logged in else a new user account will be created. You can then use additionalUserInfo.isNewUser property from the result received from signInWithPopUp method to check if the user is new.

firebase.auth().signInWithPopup(provider).then(function (result) {
  const {additionalUserInfo: {isNewUser}} = result;

  console.log(isNewUser ? "This user just registered" : "Existing User")
})

For the new Modular SDK (V9.0.0+), the same can be written as:

import { signInWithPopup, getAdditionalUserInfo } from "firebase/auth"

const result = await signInWithPopup(auth, googleProvider);

// Pass the UserCredential   
const { isNewUser } = getAdditionalUserInfo(result)    

So far, as I understood, you have two options to log in to your website: one is to make a local username/password account on your website, and the other option is to use your google account. I suppose the best way would be to check if the Google account is linked to any existing user using additionalUserInfo.isNewUser, and then linking up your google account with the account that is created via your website locally by following this article: https://firebase.google.com/docs/auth/web/account-linking?hl=lt

Once you have Firebase dependency inside your application. You can use createUserWithEmailAndPassword method to do that.

firebase
.auth()
.createUserWithEmailAndPassword("email@domain.com", "123123")
.then(data => {
    data.user.updateProfile({
        displayName: this.form.name
    }).then(() => {});
})
.catch(err => {
    this.error = err.message;
});