I wrote "I eat a second cookie", I would have accepted "I eat the second cookie"... but I don't get the "my"
In English, you can say it with "my". It's sort of like saying "this is the second cookie today for me / that I took for myself". For example, I can have my first/usual coffee in the morning. Then later I can have my second coffee (for that day). It's basically another way of saying I had one coffee in the morning and one later on. At other times this can relate to watching how much you consume or having a presumed maximum in your mind that you don't want to/shouldn't go over. Let's say you're allowed five cookies each day maximum (you're on a diet), and you often eat cookies. Then if you say "It is my second cookie" it's like a reminder to self how many you had and how many more you are left to take. Conversely, someone can say to emphasise how few they took "But I only (still) take my (allowed) second cookie (for the day)!" I don't know if this makes much sense - hope works :)