Yes, I care. And how do I fight back? According to my USA friends in the know, someone having a copy of my USA passport, can do very little damage. But someone having a copy of my license can cause many issues. In my recent month long trip through Asia, South Pacific, and a quick stop in Europe, my passport was either scanned or copied at every lodging. I'm not an expert on the rules in Japan and Norway, but self check-in kiosks were used in my hotels - scanning the passport does the initial step. So there's that.
I'm well travelled in the US. I've only had three stays where a copy of the license was required. One was a HIEx off of the NJ Garden State Pkwy, about 6-7 years ago. The reservation confirmation clearly indicated that a copy of my license would be taken at check-in. One of my colleagues challenged the front desk agent on the requirement. She got the manager. It was a requirement of the township due to regular criminal activity in the area apparently (the township code was clearly posted). However, the copy was made an placed in a lock box. The manager explained that the copies were destroyed after several days - right to the shredder. We were OK with that.
I stayed at a lower end chain property in So. VA last year - part of a road trip and I just needed a bed. Obviously an ID is needed to check-in. My policy is to show the agent my ID, while holding it in my hand. He tried to grab it from me and was not successful. I asked why he needed to hold it. He said to make a copy for the government. My answer to that was "please provide documentation of the government requirement". He backed down. (I know the area very well and pass through for a night or two every year - I knew it was not a requirement.)
The 3rd time was also last year at a HIEx near JFK. Handwritten signage posted on bullet proof glass at the check-in area indicating that a copy of a photo ID would be made without exception at check-in. I was traveling internationally and offered my passport. "Why no license", I was asked, more than once. He was clearly irritated that I would not give him my license. The passport sufficed. (And if it hadn't been February and still daylight, I'd have found another property.)
For the US, don't let them handle your license or passport - just show it to them. Again, for the US, if the agent insists on a copy, have the agent produce documentation that requires a copy be made and what the circumstances are, i.e. a local ordinance. I think it is extremely rare that a US property requires a copy of anything.
I've looked for an information resource that is all inclusive about rules for check-in and identification throughout the world, but have failed. And with my legitimate NJ example, the rules will vary anyway country by country, region by region, etc.
Hundreds of copies of my passport have been made over 30 years and I just don't worry about it outside of the US.