I think my point would be that, by discussing the minute semantic / philosophical points of the concept of "identity", you're
still letting them frame the discussion that way. It's a word that they choose to describe something which it isn't. First is to just not go along with it, not to dig in and try to beat them on their own territory (if you succeed, you won nothing).
For the same reason I won't go into discussions about the finer moral points when stealing is wrong or not, if the topic is copyright. Especially not get carried into far-fetched analogies such that it is okay if a starving family steals the blueprint for a 3D printed load of bread or whatever.
In that sense, the term "intellectual property" is actually similarly problematic as "identity theft". While it evokes the connotation of "property", intellectual_property is actually just a legal term that stands on its own and derives nothing from the common concept of "property" except where explicitly defined as such.
Except that identity_theft is, afaik, not a legal term. I believe it stems from the idea of the loss of an interconnected number of (mostly electronic) credentials, an adversary could use to, in a sense "become you", and wreck one's life. This then became a serious fear, that was (in the public) not quite blamed on terrible security practices of powerful entities, but on the ever-growing interconnectedness and electronicification of all aspects of our life. In fact literally about the fear that the large amount of data about us in these computer databases, would some day mistaken to be us and identify, regardless of its truth in the real world. But identify_theft has always been painted as a sort of "curse of the modern age", our penance for living in an ever automated society, kind of typical Hollywood morality story.
Except these credit companies seem to be just focusing on the "wreck your life" part, twisting the definition around, that suddenly a security failure with their authentication/credential system gets to be blamed on the general societal menace of identity_theft, mainly because their error has the capability to wreck one's life.
I'm pretty sure Baudrillard or some other person in critical theory / semiotics has written some interesting stuff about this. Now that is a philosophical discussion on this topic that I would actually find worthwhile.