Identity means that:
1. We have some EQ function such that if EQ(X, Y) reports true, then X and Y are the same object; and
2. There doesn't exist any function which is more discerning than EQ; if EQ concludes that X and Y are the same, then no other function can find a difference.
In programming, the currrent object may be different from that object moments ago because of mutations. However, inside the virtual machine, we are forced conclude it's the same object. The reason is that within the machine, we do not have a function which can distinguish the two objects. And the reason for that is that under mutation, we we do not simultaneously have the previous object and the new object. Therefore, we cannot apply both of them to a function.
We can discuss differences between yesterday's you and today's you, and find differences. But the discussions of you are not you; they are ideas of you.
Likewise in the machine, we can snapshot the state of an object into some serialized format or whatever, and then two snapshots at different times are different, informing us how the object changed. The two snapshots are not EQ, but they are not the object.