This is an excellent point. When a cell A is not dividing, we want to say that A stays the same cell even as it moves and changes over time. When A does divide into two parts, we have no way to decide which part is the original A. If one part is on the left and the other part is on the right, we may equally well give two versions of events: "A split off its right half and moved to the left," or "A split off its left half and moved to the right." It's like asking if a zebra is black with white stripes or white with black stripes.
Leibniz's law of identity has very little to say about any of this. Even when a cell is not dividing, but merely changing over time, Leibniz's law of identity does not allow us to say it is "the same" cell, as its properties are different. Nor does Leibniz's law of identity allow you to say you are the same person you were this morning.
We might use Leibniz's law in a different way, in which we speak of the whole history or timeline of an object, extending into the past or future, as having a single identity. You are not identical with your self this morning, but you in the present and you in the morning share the same timeline. But this does have difficulty accounting for the cell's mitosis, as we have no rule or principle to tell us which half of the divided cell gets the "original" timeline. The timeline apparently branches; we have a free choice of which to call the original.