Thank you, that was very helpful.
Okay, so the issue was, that the menu item is inside an anchor tag, which has to have “href” attribute (where it points to), and in this case it was the shortcode name, which was not a valid URL.
The only way I was able to trick the menu to think it is a menu link is, that I have changed the URL to “#search” – which is a non-existent anchor. Now when clicking, it updates the address bar to this anchor, but does not reload the page nor does anything else.
Another option would be a custom code to simply disable the link, but this way no coding is required.
I have also adjusted the search position to “absolute” from the default “static”, and changed the top and the right positions to match the menu bar.