I noticed your post yesterday, and have waited for someone to respond with knowledge. That hasn't happened yet, so I'll hazard a guess.
I did a quick google search which led to a bunch of nonsense (perhaps you did the same), and that leads me to think maybe it's a marketing phrase and means nothing. I used to work in kitchen design and, when I would meet some new colleague, they would always describe their work as "high-end". No one ever said, "Oh, we do some pretty mundane stuff!"
That said, there certainly are organizations that rate banks according to quantity of their assets, perceived stability, etc. so maybe it is an industry term. Some context might help. Where did you see this phrase?
I have not heard of the term. There are first class service, first class flights, first class hotels but it is strange to think of banks that way. There are big and small banks with first class service but I don't know what first class bank entails.