Hi,
Thank you for the details!
Some special themes may have custom made search bars, which the theme author may program not be be replacable automatically. In those cases the only possible way is to either programmatically replace or use the plugin shortcode.
In your case the theme search bar redirects to a custom made search results page, which I guess is specifically made for the integrated theme search. These custom query pages usually does not work with other search plugins. In your case the WordPress search results page is this: https://advisortattoo.it/?s=uno
That is a bit different from what the homepage custom search integrates to.