How to add template suggestions for form elements of a certain form ID?
This is kind of tricky as you'd have to write a lot of custom code to pass the form id to a form element as you stated above.  Fortunately, there is a contrib module called &quot;Themable forms&quot; which does exactly that, both for form elements and form element labels.
So for example, if I am on the user login page, for the form elements, I see these default theme hooks for all individual form fields.
&lt;!-- THEME HOOK: 'form_element' --&gt;
&lt;!-- BEGIN OUTPUT from 'themes/custom/mytheme/templates/form/form-element.html.twig' --&gt;

&lt;!-- THEME HOOK: 'form_element_label' --&gt;
&lt;!-- BEGIN OUTPUT from 'themes/custom/mytheme/templates/form/form-element-label.html.twig' --&gt;

Note, there is no granularity at all for the above.
Now, after I enable Themable forms, I start to see very granular theme hook suggestions being output in the form of
  form-element--[form-id]--[element-type].html.twig
  form-element--form-id--[form-id].html.twig
  form-element--type--[element-type].html.twig
  form-element.html.twig

So for example, with the username field, I now see
&lt;!-- THEME HOOK: 'form_element' --&gt;
&lt;!-- FILE NAME SUGGESTIONS:
   * form-element--user-login-form--textfield.html.twig
   * form-element--form-id--user-login-form.html.twig
   * form-element--type--textfield.html.twig
   x form-element.html.twig
--&gt;

&lt;!-- THEME HOOK: 'form_element_label' --&gt;
&lt;!-- FILE NAME SUGGESTIONS:
   * form-element-label--user-login-form--textfield.html.twig
   * form-element-label--form-id--user-login-form.html.twig
   * form-element-label--type--textfield.html.twig
   x form-element-label.html.twig
--&gt;

You'll now note a few things, the form id in the form element suggestion as well as the name of the form element itself.
This should get you where you want to be according to your question above.
You can install the module with:
composer require drupal/themable_forms