There are many dictionary definitions of identity. Take Merriam-Webster:

"1a: the distinguishing character or personality of an individual

2: the condition of being the same with something described or asserted

3a: sameness of essential or generic character in different instances"

Or the Cambridge dictionary:

"a person's name and other facts about who they are:"

Of course, you're always welcome to intentionally pick the incorrect context (going back to Merriam-Webster):

"4: an equation that is satisfied for all values of the symbols"

And just as easily rant the name of the crime has nothing to do with math so it needs to be renamed.

If it's open to interpretation, you're choosing the worst interpretation.

How this is interpreted is the primary problem that needs to be solved in order to legislate the issue, and from the interpretations you're supporting, it seems you're firmly on the side of the parties failing to identify fraudsters correctly, versus the innocent individuals.

"Worst interpretation" isn't about matter of preference on which interpretation is moral rather intentionally choosing the least supportive definition as proof the phrase is a misnomer. Ignoring the Cambridge one, which seems to be post phrase, definitions like "sameness of essential or generic character in different instances" are very stealable things, even if another definition is not.

As I've stated I'm in favor of the individuals. When doing so, I like to stick to factual and logical reasons why instead of insist the root of the problem is the chosen definition of "identity" does not match my preferred one. The root of the problem is the current process for dealing with identity theft is more burdensome than it needs to be on individuals. If you change the name it's still too burdensome because, outside of you, everyone already interprets "identity" to mean what it should in this case. That the process is bad is not proof the interpretation of the phrase is bad and needs to be fixed, it's proof the process is bad and needs to be fixed.

Just because someone disagrees with one component of your stance does not mean they automatically agree with all opposing positions that followed from it.