first of all, what versions of wppizza and wpml are we talking about please?

Assuming you are happy to edit some php files (let me know if that is not the case )
could you please try the following

(I dont use WPML really as such , so it would be great if you could let me know if the below solves your problem and does not cause any others elsewhere before I makes this actually part of an update)

in [your site path]/wp-content/plugins/wppizza/classes/class.wppizza.categories.php you will find at approx line 118 the following codeblock

if(function_exists('icl_get_languages')){
global $sitepress;
$languages = icl_get_languages();
/**loop through non current languages to get categories**/
foreach($languages as $lang_code=>$lang_arr){
if($lang_code!=ICL_LANGUAGE_CODE){
$sitepress->switch_lang($lang_code);
/*get cats for that language**/
$wpml_sort_cats_lang[$lang_code] = get_categories($args);
}
}
/**make sure we switch back to current language when done**/
$sitepress->switch_lang(ICL_LANGUAGE_CODE);
}

REPLACE that whole codeblock with

if(function_exists('icl_get_languages')){
global $sitepress;
$languages = icl_get_languages();
/**loop through non current languages to get categories**/
foreach($languages as $lang_code=>$lang_arr){
if($lang_code!=ICL_LANGUAGE_CODE){
$sitepress->switch_lang($lang_code);
/*get cats for that language**/
$wpml_sort_cats_lang[$lang_code] = get_categories($args);
}
/**make sure we switch back to current language when done**/
$sitepress->switch_lang(ICL_LANGUAGE_CODE);				
}
}

i.e move the second ->switch_lang() inside the loop
let me know if that solves the issue in your case too – seems to work fine when i do this here in a test environment

when you are saying “only a part of it”
which “parts” in particular do not work…
(admin? frontend? some global text strings ? or particular plugins/themes only ?)

i’m a bit confused now
if I go here:
https://www.[yoursite].it/wp-admin/plugins.php
i get different languages in the descriptions (quite probably because the plugins and their descriptions have no IT translation)
even though WPPizza is DE-activated ???
So i assume you are not actually referring to this ?!

if i look at my clone of your site here now i get the same plugins names/descriptions (/wp-admin/plugins.php) as well as the same admin labelling – i.e the stuff on the left like “Media”, “Pagine”, “Prodotti” “Openig Hours” etc etc (/wp-admin)
regardless of whether wppizza is active or not (i can show you screenshots if you like)

i must be missing something…:(
please can you give me a distinct example (some text/heading/label somewhere) that changes/is different
depending on whether wppizza is activated or not ….

thanks for the screenshots. they help a lot and i think i know where the issue is now (just need to find a few variables that are set by WP/WPML)
will post back here when I think i solved it