It's per account and can be doubled with a "payable on death" designee.
I think you're mistaken about the first part of your statement:
The FDIC says, "All single accounts owned by the same person at the same bank are added together and insured up to $250,000." (see https://www.fdic.gov/deposit/covered/categories.html )
But you are correct that it can be doubled, tripled, quadrupled, or even greater by opening different account types such as a Retirement Account, a Joint Account, a Revocable Trust Account, etc., at the same bank since each account type seems to get separate treatment (according to the same FDIC link).