how to make react-hook-form work with multiple forms in one page?

Welcome to StackOverflow @webcoder
You are using the same hook instance for both forms. You will have to reuse with different names.

import React from "react";
import ReactDOM from "react-dom";
import { useForm } from "react-hook-form";

import "./styles.css";

function App() {
  const {
    register,
    formState: { errors },
    handleSubmit,
  } = useForm({
    mode: "onBlur",
  });

  const {
    register: register2,
    formState: { errors: errors2 },
    handleSubmit: handleSubmit2,
  } = useForm({
    mode: "onBlur",
  });

  const onSubmit = (data) => {
    alert(JSON.stringify(data));
  };

  const onSubmitEmail = (data) => {
    alert(JSON.stringify(data));
  };

  return (
    <div className="App">
      <form key={1} onSubmit={handleSubmit(onSubmit)}>
        <div>
          <label htmlFor="firstName">First Name</label>
          <input
            name="firstName"
            placeholder="bill"
            ref={register({ required: true })}
          />
          {errors.firstName && <p>This is required</p>}
        </div>

        <div>
          <label htmlFor="lastName">Last Name</label>
          <input
            name="lastName"
            placeholder="luo"
            ref={register({ required: true })}
          />
          {errors.lastName && <p>This is required</p>}
        </div>
        <input type="submit" />
      </form>

      <form key={2} onSubmit={handleSubmit2(onSubmitEmail)}>
        <div>
          <label htmlFor="email" placeholder="bluebill1049@hotmail.com">
            Email
          </label>
          <input name="email" ref={register2({ required: true })} />
          {errors2.email && <p>This is required</p>}
        </div>
        <input type="submit" />
      </form>
    </div>
  );
}

const rootElement = document.getElementById("root");
ReactDOM.render(<App />, rootElement);