It's certainly possible, but it's unlikely. The Lyft driver's own assets are likely small, and most lawyers suing the driver would rather settle for an amount within the $1 million insurance policy, rather than risk getting nothing by going to court to try to get more than the insurance policy.
Even many doctors don't have malpractice insurance above $1 million (depending on speciality and state) -- most policies are $100K-$300K/claim, $1MM-$3MM/all claims (http://jop.ascopubs.org/content/3/5/274.full). So Lyft is providing more liability insurance than many doctors carry.
If you're demanding absolute 0% risk for the Lyft driver financially, this would not cut it, but very little anywhere is absolutely no risk.
