var sel1 = $("#se1");
var sel2 = $("#se2");
function swap() {
let selection1 = sel1[0].selectedIndex; // Store the first list's selection
sel1[0].selectedIndex = sel2[0].selectedIndex;
sel2[0].selectedIndex = selection1;
// Test
console.log(sel1.val(), sel2.val());
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
<select id="se1">
<option value="1">DOG</option>
<option value="2">CAT</option>
<option value="3">BIRD</option>
</select>
<button onclick="swap()">SWAP</button>
<select id="se2">
<option value="4">DOG</option>
<option value="5">CAT</option>
<option value="6">BIRD</option>
</select>