The person or the “I” is not, on one account, a single “thinglike being” but a “unity of experiences“ existing over time which does not necessarily follow or adhere to a discernible logic.
So, in some sense, when we refer to ourselves we are referring to a changing being, a changing consciousness unified by a body and name (and maybe a soul…if you believe in that…I do…although I don’t think Heidegger does).