Create subpage /user/ or /my-profile/ like /author/ with additional query like /user/user123

If I understand you well, you can do that with add_rewrite_rule(); function to map the url to specific query variables. https://codex.wordpress.org/Rewrite_API/add_rewrite_rule

function myfunc_rewrite_rules() {
  add_rewrite_rule('user/?([^/]*)', 'index.php?pagename=user&username=$matches[1]', 'top');
 }

 function myfunc_username_query_vars($vars) {
  $vars[] = 'username';
  return $vars;
 }

add_action('init', 'myfunc_rewrite_rules');
 //add query vars (username) to wordpress
 add_filter('query_vars', 'myfunc_username_query_vars');

function myfunc_display() {
  $userpage = get_query_var('pagename');
  $username = get_query_var('username');
  if ('user' == $userpage && '' == $username ):
   //show all users
   exit;
  elseif ('user' == $userpage && '' != $username ):
   //show specific user
   exit;
  endif;
 }

 //register plugin custom pages display
 add_filter('template_redirect', 'myfunc_display');

This tutorial can help: http://clivern.com/how-to-add-custom-rewrite-rules-in-wordpress/

UPDATE:

After you run this code, you must go to Settings->Permalinks and just hit save button to flush all rewrite rules.(This is the solution for many permalink issues)