[DivMan]

Так?

var listCity = document.querySelector(".styled-select");
	var showListCity = document.querySelector(".chooce-option");
	var cityValue = document.querySelector(".chooce-value");
	
	var options = document.querySelectorAll("li");
	
	
	showListCity.addEventListener("click", function(e){
		if(listCity.style.display != 'block') {
			listCity.style.display = 'block';
		}
		
		else {
			listCity.style.display = 'none';
		}
		
		
		
	})
  
        options.forEach(function(item, i, arr) {
	    options[i].addEventListener('click', function(t){
		console.log(t.target.getAttribute("data-value"))
		showListCity.value = t.target.innerHTML;
				
		cityValue.value = t.target.getAttribute("data-value");
		console.log(cityValue.value)
		listCity.style.display = 'none';
	})
});