Try choosing the searchable layout for the select box: https://i.imgur.com/z5EV8X7.png
That will trigger the same script and layout for the select box.
I tried to do that, but this only works with custom fields, and We need to make it work for category & taxonomy fields.
But don’t worry, I solved the issue, just using the dropdown with search (that has the same style as the multiselector) and then with CSS I removed the inner search bar adding the display as none in the specific class.
Anyway, your reply solved or not my problem, thank you for your support!