The damage, in most cases, is to your credit reputation and the time and effort it takes to fix the problem. It is generally a major inconvenience as it takes a whole lot of effort to clean up the mess and some lingers for years after. In the worst cases, a victim of identity theft could be arrested and jailed until they were able to prove that that they were a victim, which takes time and effort.
My father was a victim in the months before he passed away. It started innocently with small charges on his Discover card that he hadn't made. The thief then changed his mailing address for the account to someone else's address -- not the thief -- and then opened several new accounts using my father's identity. The next step was to get a cash advance and transfer money between accounts before ultimately attempting to withdraw it. I can't recall if that last part was successful, but I spent hours and hours on the phone with banks and credit card companies, had to document everything that was fraudulent, and I froze his credit reports to prevent new credit from being issued. Ultimately, he was not responsible for a penny of the fraud, but it was a massive pain to deal with. And, the thief was likely never caught.
Adding to that, while he was not responsible for the thousands of dollars, we all pay for this because the banks and credit card companies need to account for the losses in their cost of doing business which drives up interest rates and fees.
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stephen boots Microsoft MVP 2004 - 2020
"Life's always an adventure with computers!" 
