JSFiddle - React, Tailwind, and code Playground

HTML

<select id="first">
    <option value="none">None</option>
    <option value="one">Toyota</option>
    <option value="two">Nissan</option>
    <option value="three">Dodge</option>
</select>

<select id="second">
    <option value="none">None</option>
    <option value="one">Toyota</option>
    <option value="two">Nissan</option>
    <option value="three">Dodge</option>
</select>

JavaScript

$("#first").change(function() {
    if($("#first").val() != "none") {
        $("#second option[value!="+$("#first").val()+"]").show();
        $("#second option[value="+$("#first").val()+"]").hide();
    } else {
        $("#second option[value!="+$("#first").val()+"]").show();
    }
    $("#second").val('none');
});

$("#second").change(function() {
    if($("#second").val() != "none") {
        $("#first option[value!="+$("#second").val()+"]").show();
        $("#first option[value="+$("#second").val()+"]").hide();
    } else {
        $("#first option[value!="+$("#second").val()+"]").show();
    }
    $("#first").val('none');
});