A philosophical answer coming: Mine, yours...
What do they mean? They mean that a human has something, possess something, they indicate possession.
What you said in your question is correct but as indicated in a comment they are rarely used, the reason is objects cannot possess things. I see a computer with a keyboard, I still call it a computer. I see bread with meat in it, I call it hamburger.
You see, when you modify a property of an object, you either keep calling it by the original name or create a new name. Object cannot possess objects, you're going to end with an object.
Animals cannot possess objects, they cannot own things. I hope I'm being clear here. People started saying "it's mine" to indicate what they bought etc. Many years later language changed and we started using "his" for dogs or whatever.
My teacher used to hate referring to animals by he/she. But again grammatically what you said is correct.