Вы могли бы просто сделать
option:checked {
background-color:#0F0;
}
<select id="category" required>
<option value="">Please select</option>
<option value="Cats" selected class="active">Cats</option>
<option value="Dogs">Dogs</option>
<option value="Birds">Birds</option>
</select>
ИЛИ
$('#category').change(function() {
$('option', this).removeClass("active");
$('option:checked', this).addClass("active");
});
option.active {
background-color:#0F0;
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<select id="category" required>
<option value="">Please select</option>
<option value="Cats" selected class="active">Cats</option>
<option value="Dogs">Dogs</option>
<option value="Birds">Birds</option>
</select>