Flat out wrong. They nearly always are. They frequently have their credit impacted as a result of non-payments for credit, mortgages, etc. Also, I see "identity theft" as a subset of theft by conversion: https://definitions.uslegal.com/t/theft-by-conversion/ "Theft by conversion occurs when someone wrongfully uses property or funds of another for their own purposes." A person's name, SSN, address, financial history, credit rating, etc. are all their property. And it is being used without their consent and in blatant violation of other laws that already exist. They clearly are a victim (but not the only victim) of these illegal acts.
Also, let's not forget that it typically takes an enormous amount of time and effort to fully undo the damage. That time alone has a cost - and it has been forcefully taken from the person whose identity has been stolen by the perpetrator(s) of these crimes.