Save your pet name in a tag, or a note since those exist for customers. pet_name obviously is not a customer field.
<p>
<label>Pet Name:</label>
<input type="text" value="" name="customer[note]" class="input-text-1" />
<input type="radio" name="customer[tags]" value="cat">Cat
<input type="radio" name="customer[tags]" value="dog">Dog
</p>