Common ancestry of our cells to the first unicellular cell
What it seems you're struggling with here is a lack of clear definitions.
Yes, the unity of core cellular mechanisms implies an unbroken line of descent back to the origins of life. But that does not in any way mean the cells are actually the same. I think that you would find it instructive to learn about the puzzle of the Ship of Theseus, which is the essence of the issue you are dealing with. In short, once changes can happen (e.g., mutation, as you note), it's possible to have both continuity and a fundamentally different item.
Beyond this, it's also worth noting that at every step we are dealing not with individuals but populations, even likely back into the murk of abiogenesis (when the populations would be various chemicals and protocells of some sort), and that genomes are also modified in a much wilder variety of ways, e.g., all the various forms of horizontal gene transfer.