As a possessive pronoun, what does "mình" exactly denotes? In the first place, I translated this sentence as "He finds my mouse".
To avoid repeating word "anh ấy", "cô ấy" in the same sentence once more time as a possessive pronoun, you can use the word "mình" instead.
"He finds his mouse". Instead of translating: "Anh ấy tìm con chuột của anh ấy, you can say: "Anh ấy tìm con chuột của mình."
"She drinks her water." = Cô ấy uống nước của cô ấy. You can say: "Cô ấy uống nước của mình. "
"I eat my bread" = Tôi ăn bánh mì của tôi. You can say: "Tôi ăn bánh mì của mình." :)
No, you can't say the same way.
As you can see in three examples above, each sentence just talks about one person, includes possessive of that person (he -his, she-her, I - my), mình is just used in that case.
But in this case, you say: "He finds my mouse", it talks about two people. The first one, it directly talks about "he" and other one, indirectly talks about "I" (possessive of " I " is my), so you have to say clearly: "Anh ấy tìm con chuột của tôi" to avoid confusing. :)
I think the problem comes from "mình" being what is often called a false friend in language. It sounds close to the English word "mine" and sometimes means exactly that. I find it easier to think of "của" as "belongs to" and "mình" as referering to the preceding pronoun, taking care to remember that it only becomes "mine" if preceded by tôi.