Pet owner is asking a picture of my driving license?

I would refuse. Telling her that you are uncomfortable with it. your drivers license number and home address is personal information that client doesn't need nor should they have it. Your reviews speak for themselves. I've learned clients like that are typically hard to work with and make unreasonable demands later on.

I'm curious, what sensitive information is on your driver's license?

I can understand and have always been reluctant to provide my driver's license to places that I felt had no need for it. Tell the client that you have been vetted by Rover and they should look at all your "verified" reviews, which means they have book their services through Rover.

Tell them that you are not comfortable due to the proliferation of privacy and identity theft issues these days. They should be able to rely on the posted reviews. However, if they have additional questions they should call Rover Customer Support at [Edit: Rover’s contact options have changed. Visit the Rover Help Center at https://support.rover.com/ to find the phone number, help articles, or chat with the team] to discuss their background checking process.