Nope. They ask for year/make/model, ask if it's modified, ask if it gets parked outside or in a garage, ask for money, print my cards, and thank me for my business.
It's been awhile since I needed to make a claim... but I dont even remember ever needing to prove ownership during claims processing. Relatively sure they just sent their guy out to take pictures, he asked where I wanted to send it to get fixed, and told me I was good to go.
For the longest time my fiances parents had her car on their policy, they definitely didn't own it. Insurance companies don't care, they just wanna get paid.