The problem that you face is that demo you show actually copies the original select form element and replaces it with a DIV based structure. There are a number of steps you need to take to go down that route.
Firstly, go to Wp admin -> Search and Filter -> Settings and change the Combobox script to select2.
Then, you need to go to the Select2 documentation and apply Select2 to each of the fields in the form (use the class sf-input-select to target them). I think that the jquery would look like this:
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