It looks like in your example the Child -component is not rendered at all.

If your plan is to show the Child-component after you have clicked the "Click me" -button, then you could render it conditionally and pass in the data directly without using the context-api. Like this:

    {showChild && 
        <Child data={{name, email, password}}></Child>
    }

You need to add state -variable showChild and set it true in the handleClick -function.

Below is a working example:

    function Parent() {
      const [name, setName] = useState("");
      const [email, setEmail] = useState("");
      const [password, setPassword] = useState("");
      const [showChild, setShowChild] = useState(false);
    
      const handleClick = () => {
        setShowChild(true);
      };

      return (
        <div>
          <input type="text" onChange={(e) => setName(e.target.value)} />
          <br />
          <input type="text" onChange={(e) => setEmail(e.target.value)} />
          <br />
          <input type="text" onChange={(e) => setPassword(e.target.value)} />
          <br />
          <button onClick={handleClick}>Click me</button>
          {showChild && 
            <Child data={{name,  email, password}}></Child>
          }
        </div>
      );
    }

    const Child = (props) => {
      const data = props.data;
      return (
        <>
          <div>{JSON.stringify(data)}</div>
        </>
      );
    };
    export default Parent;

If you really have to use the Context-api, then you need to wrap the child-component to to a FormContext.Provider and pass obj as a value property. Like this:

       <FormContext.Provider value={obj}>
          <Child></Child>
        </FormContext.Provider>

Below is a working example:

    function Parent() {
      const [name, setName] = useState("");
      const [email, setEmail] = useState("");
      const [password, setPassword] = useState("");
      const [showChild, setShowChild] = useState(false);
      const obj = { name, email, password };
    
      const handleClick = () => {
        setShowChild(true);
      };
    
      return (
          <div>
            <input type="text" onChange={(e) => setName(e.target.value)} />
            <br />
            <input type="text" onChange={(e) => setEmail(e.target.value)} />
            <br />
            <input type="text" onChange={(e) => setPassword(e.target.value)} />
            <br />
            <button onClick={handleClick}>Click me</button>
            {showChild &&
           <FormContext.Provider value={obj}>
              <Child></Child>
            </FormContext.Provider>
            }
          </div>
      );
    }
    const Child = () => {
      const data = useContext(FormContext);
      return (
        <>
          <div>{JSON.stringify(data)}</div>
        </>
      );
    };
    export default Parent;

Try this . <FormContext.Provider value={{obj}}> <Child /> </FormContext.Provider>;