When you create the PlaceAutocompleteElement object, you have access to the seach input element and the dropdown element with the search results.

So you can append those elements to your HTML. Once they are in your HTML, you can give them custom classes to style.

Here is how:

const container = document.querySelector(".container"); //An example container in your current DOM
const placeAutocompleteElement = new google.maps.places.PlaceAutocompleteElement();
const inputAutocompleteElement = placeAutocompleteElement.Eg; //The  search input autocomplete element
const dropdownElement = placeAutocompleteElementjg; //The dropdown with search results

//Now append both elements to your DOM
container.append(inputAutocompleteElement);
container.append(dropdownElement);

//Give the elements your custom styles
inputAutocompleteElement.classList.add("search-input");
dropdownElement.classList.add("dropdown-results");