Literally all std::unique_ptr does is wrap the pointer. If we make two unique_ptrs which both think they own object X then not only will they both exist and point at X (so they don't actually ensure they're unique) they'll also both try to destroy X when they disenage e.g. at the end of a scope, leading to a double free or similar.
Uniqueness is a property you are promising, not one they're granting.
I expect you knew this, but it's worth underscoring that there is no magic here. This really is doing only the very simplest thing you could have imagined, it's a pointer but as an object, a "smart pointer" mostly in the same sense that if I add a cheap WiFi chip to this refrigerator and double the price now it's a "smart fridge".