Adding fields to the “Add New User” but the form data not saved into DB

I’ve found these errors in your code:

• The saving proccess of custom user fields is hooked to registration but not to update user profile actions.
• You are using update_usermeta, a obsolete function, use update_user_meta instead.
• You are using get_the_author_meta with $user->ID without checking before if $user is an object, which generate errors on new user registration form (also I recommend use get_user_meta instead).
• You are not sanitze/validate the data of company field (I’m not sure what type of data you expect here, propposed saniteze_text_field as example).

I’ve made some changes and tested.

function custom_user_profile_fields($user){
    $previous_value="";
    if( is_object($user) && isset($user->ID) ) {
        $previous_value = get_user_meta( $user->ID, 'company', true );
    }
    ?>
    <h3>Extra profile information</h3>
    <table class="form-table">
        <tr>
            <th><label for="company">Company Name</label></th>
            <td>
                <input type="text" class="regular-text" name="company" value="<?php echo esc_attr( $previous_value ); ?>" id="company" /><br />
                <span class="description">Where are you?</span>
            </td>
        </tr>
    </table>
<?php
}
add_action( 'show_user_profile', 'custom_user_profile_fields' );
add_action( 'edit_user_profile', 'custom_user_profile_fields' );
add_action( "user_new_form", "custom_user_profile_fields" );

function save_custom_user_profile_fields($user_id){

    if(!current_user_can('manage_options'))
        return false;

    # save my custom field
    if( isset($_POST['company']) ) {
        update_user_meta( $user_id, 'company', sanitize_text_field( $_POST['company'] ) );
    } else {
        //Delete the company field if $_POST['company'] is not set
        delete_user_meta( $user_id, 'company', $meta_value );
    }
}
add_action('user_register', 'save_custom_user_profile_fields');
add_action( 'personal_options_update', 'save_custom_user_profile_fields' );
add_action( 'edit_user_profile_update', 'save_custom_user_profile_fields' );

Also, this piece of code block any user to save the data if he/she has not “manage_options” capability, which is a capability usually only administrators have, so users won’t can update these fields:

if(!current_user_can('manage_options'))
    return false;

So, remove it or make another checking of user capabality if it is needed. For example, checking if current user can edit the user being update seems better:

if ( !current_user_can( 'edit_user', $user_id ) )
    return false;