Create a post in custom post type using field in registration form after users submit form

First, add the custom field to the registration form:

add_action( 'register_form', function(){
    ?>
    <p>
        <label for="institute">Institute</label>
        <input type="text" name="institute" class="input" value="" size="25">
    </p>
    <?php
} );

Then, hook to user_register action which gets called after a user has been registered succesfully.

add_action('user_register',function($user_id){
   $my_institute = array (
     'post_type' => 'Institute',
     'post_title' => $_POST['institute'], // Custom field value
     'post_content' => 'This description is not needed',
     'post_status' => 'publish',
    );
    wp_insert_post( $my_institute );
});