Couple thoughts:

1) In humans, a computer seems a better analogy, with the parts being the body and the data being the consciousness. You could systematically replace every part (cell) and you'd still end up with that person/data.

2) This is coupled (and complicated) with the notion of definitions, and that regardless of whether parts-swapping even takes place at all, the "me-ness" of me is subjective, and the definition changes from person to person. Other people don't agree where I begin and end, and other people hold different definitions with regards to themselves. The definitions themselves are fluid. A person with Alzheimer's ceases to be in many people's eyes, while a person with all of their mental faculties but sever limb loss might more often be considered intact as a person with a physical caveat. I think in the aggregate, the balance of what constitutes a person leans towards the consciousness of that person according to most.

3) Our subjective definitions are formed in a pool of ignorance and altered perceptions. I don't mean we're stupid, -just much of the talk about what makes a person a person fails to bear in mind the fact that a human being is really a colony of innumerable life forms working together, and could not exists without the group of them working together. It's hard to draw the line between what is symbiotic and what is native in many cases. This also nudges us towards consciousness being the unit of the self, as it can reside on a conglomerate or individual unit of life and it wouldn't matter in a sense.

For me, the ship is still the ship if every board is replaced, as long as they are not all replaced so quickly that its observers do not have enough time to assign/upkeep the same identity to it in the interim, -a process which will yield different results per person, and in many ways is reached by consensus.