Admin submenu issue with PHP not detecting two strings as equal

string(77) "edit-tags.php?taxonomy=acs-field-group-category&post_type=acs-field-group"
string(73) "edit-tags.php?taxonomy=acs-field-group-category&post_type=acs-field-group"

Same string, different length!

No, they’re not the same string — if you had viewed the raw/server-generated HTML source, or used this instead of the var_dump($submenu_file);:

echo '<pre>'; var_dump($submenu_file, $sub_item[2]); echo '</pre>';

Then you would have seen the correct string which uses &amp; instead of just &:

string(77) "edit-tags.php?taxonomy=acs-field-group-category&amp;post_type=acs-field-group"

So that means, your (sub-)menu slug (the 5th parameter for add_submenu_page()) needs to use the ampersand’s HTML entity, i.e. &amp;, like so:

add_submenu_page( $slug, __('Categories','acs'), __('Categories','acs'), $cap, // wrapped for brevity
    'edit-tags.php?taxonomy=acs-field-group-category&amp;post_type=acs-field-group' ); // ← use &amp;

That way, the submenu would be active when the submenu page is loaded.

And actually, you can see a core example here in wp-admin/edit-tags.php (WordPress v6.0.1), which implies that $submenu_file (the submenu slug) expects &amp; to be used.

PS: Remember to restore the original wp-admin/menu-header.php file — remove your “hack”/modifications.