No, they mean the private insurance company operates nationwide. There are several of them in the US

I don't think so. I think the previous poster means that most insurers serve a nationwide (or international) audience. So in terms of pooled risk, the Florida hurricanes and California wildfires aren't separate risk pools.

On your other point of comparison, we do have similar things in the states. Lots of gov programs that have "insurance" in the name, but if they ever were pooled risk insurance, they've morphed into social welfare programs a long time ago. They vary by state, and sometimes county, but they include Unemployment Insurance, Workers Compensation Insurance, Disability Insurance, Medicare, Medicaid, and even old age pension (called Social Security). Some of these, like Workers Compensation have private market competition, where others do not.

We also have agencies like FEMA that aren't insurance, but have become a sorta relief mechanism for large scale catastrophic events. We also have things like the California Earthquake Authority, which provides earthquake insurance. I have no idea if it's genuinely pooled risk or not, but it's like an order of magnitude cheaper than commercial earthquake insurance was when we had before CEA existed.

Then we have a heap of programs and agencies that provide insurance (or assurance of some sort) to industries. FDIC insures bank deposits. FNMA, FHA, et al insure (or provide liquidity) to mortgage lenders. And on and on. There are probably more forms of state run or state sponsored insurance than any one person knows. There might even be some company or program called National Insurance, but not that I'm aware of.