I think you are forgetting how html forms work, quick refresher, without all the WordPress template tags:
<form action="myaction.php" method="get">
<select name="cat" id="my_unique_select" class="my_select_class">
<option class="level-0" value="13">Category 1</option>
<option class="level-0" value="14">Category 2</option>
<option class="level-0" value="15">Category 3</option>
<option class="level-0" value="16">Category 4</option>
</select>
<input type="submit" name="submit" value="submit" />
</form>
This is how an HTML drop down form is supposed to be coded. The script “myaction.php” is expecting to be called with a GET method, and is expecting the get to be formulated as follows:
myaction.php?cat=15
The attribute “cat” comes from the name
attribute of the select, and the number 15 is the value of the selected option of the select when the submit button is pressed.
So, moving on to what you are doing (again minus all the WordPress template tags for clarity):
<form action="myaction.php" method="get">
<select name="cat" id="my_unique_select" class="my_select_class">
<option value="13">Category 1</option>
<option value="14">Category 2</option>
<option value="15">Category 3</option>
<option value="16">Category 4</option>
</select>
<select name="cat" id="my_unique_select" class="my_select_class">
<option value="23">Category 11</option>
<option value="24">Category 12</option>
<option value="25">Category 13</option>
<option value="26">Category 14</option>
</select>
<select name="cat" id="my_unique_select" class="my_select_class">
<option value="33">Category 21</option>
<option value="34">Category 22</option>
<option value="35">Category 23</option>
<option value="36">Category 24</option>
</select>
<input type="submit" name="submit" value="submit" />
</form>
lets say you have selected one option in each select. When you press submit, only the value of the last select option will be sent to “myaction.php” with a GET method. Changing the name of the selects, will result in a larger GET method, but myaction.php won’t magically know what to do with the new names, as it only understands what to do with “cat=something” in the get method. Thus what you are trying to do wont work, as you also need to modify “myaction.php” and tell it what to do with multiple select inputs.
Since “myaction.php” is a part of your program – in this case WordPress, you probably don’t want to mess with it.
That said, there are a number of enhanced search plugins available that can do boolean searches on categories. Checkout this one on the plugin page for example…