Where can I find the part number or serial number on my controller?
Updated
First, take a look on the underside of the controller, meaning the part that normally faces the floor when you are using it. Normally, there is a sticker somewhere, which contains a part number and usually the product name next to it. 4Pxxxx is the most common structure but it may start with something else. Significantly older models may instead start with "KHX-" or "HX-" alongside a number starting with "3500xx".
If you see an indent in the controller that looks like it was supposed to hold a sticker, but do not see the sticker, the sticker may have fallen off.