EMILY: You have any idea who Miles Hahn is? … He’s the president of the First National Bank. We’ve been doing business with him for years. He’s become a very dear friend of ours actually.
A name sometimes given to local banks in the US; there have been many of this name, most of which have closed down or been taken over by now. There are still examples in Florida, Utah, Nebraska, and Pennsylvania, but there’s never been one in Hartford.
It is unclear if Emily means that they use the First National Bank themselves, or if she means that Miles Hahn has been doing business with Richard through the insurance company he works for. I presume the first one.
The Annotated Gilmore Girls 