Custom page password recovery

You just need to add your own action before any errors are returned, and if there is an error, redirect back to your page and display the error

add_action( 'lostpassword_post', 'smyles123_check_for_errors' );

function smyles123_check_for_errors( $errors ){

    if ( $errors->get_error_code() ){
        // REDIRECT BACK TO YOUR PAGE TO SHOW ERRORS
        // You can probably just append the error to the URL you redirect back to, but you need to set $redirect_to value
        wp_safe_redirect( $redirect_to );
        exit();
    }

}

You could also just create your own function to handle the password reset and just copy the code from wp-login.php

There’s plenty of tutorials online to do this:
https://code.tutsplus.com/tutorials/build-a-custom-wordpress-user-flow-part-3-password-reset–cms-23811

You can also look through the code on this plugin I created a while back that handles almost everything in relation to wp-login.php … should have plenty of code information on handling custom errors, redirects, even rewrites:

https://github.com/tripflex/wp-login-flow