I think of it like a band (Could well apply to a company too) which occasionally replaces members. So long as they replace few enough members each time that people can associate it with its previous roster, people will accept it as the same band.

Even if after 20 years of incremental changes to the band none of the original members are left it will still be considered by many to be the same band (although they will all likely lament the loss of such and such member) By being recognized as the same band it inherits many of the same rights such as performing the same songs as the previous incarnations of the band (assuming of course it owns, or pays royalties to the right holders, but that is an unrelated topic).

However if too many members change at once, or they lose a member whose presence independently defined the band then people will likely not consider it the same band.

The identity is a label applied to the set, as the set changes so does the definition of the label. For the most part assuming members of the original set do not get placed in different new sets all identifying as the original label, people will generally accept the label as applying to the new set which was formed.

I would chalk it up to people being stupid, or language being imprecise. The alternative seems to be to consider every change to the set to require a new label and be identified as a new set.