Health Insurance companies will follow-up with determining if the property/land owner is responsible for the accident and try to sue. Months after ER treatment, I've had to respond that my child's cycling accident on a camp site was not because of the camp site owner's site maintenance. So it's not necessarily the victim who sues, but the victims' insurer.
I'm honestly skeptical that's a common thing however. I've certainly never encountered such a thing and, while I've seen a few cases such as yours mentioned online, it's not something I've ever heard from anyone I know.
If someone received an insurance payout (ie medical expenses covered) then there was absolutely a subrogation discussion between the insurance companies of each party. Upon being given access to electronic law records, new law students are often shocked to see their names on lawsuits they never heard about. That bicycle accident at the mall when you were a kid ... your dad's work-based health insurer sued the mall's insurer to get back the money they paid to the hospital that stitched you up. Thousands of such tiny actions are filed and settled every day.
Last time I had a cycling accident that required an ER trip, I received a packet from my insurer a few weeks later asking for details. The way the questions were phrased, they were absolutely looking for somebody else to pay. Is it common that the insurer takes the next step (actually suing another party)? No idea, but they definitely look for ways to pass the buck.
Well you'll have to define common, then. From the parent comment and "cases mentioned online" I would personally say it's common enough in the US. It's how our system works.
