clearing the cache on the tablet
Possibly, but unlikely. Browsers tend not to cache pages received over encrypted (https) connections.
They then suggested that there may be cached pages in my router
Sounds like someone was desperate for ideas. That's utter nonsense. The typical cheap routers you find in the home don't do caching .... hell, they've got hardly enough built-in memory to run their own systems let alone cache a bunch of pages !
Your ISP might run a cache server on their network....but because of the nature of encrypted connections, as a transit network your ISP can't really cache the pages anyway. Also, in this day and age, most ISPs err towards packet shaping, rate limiting and FUPs rather than caching - with the exception of running streaming nodes to cache streams on-net.
Probably to do with your tablet's acceptance of cookies.
But if your bank doesn't provide an app for your phone then you may be up a creek without a paddle trying to make your bank's system work on the typical "cut-down" browsers found on tablets. Even more so if they are the sort of company that tests their websites on whatever they have in the office (i.e most likely to be Internet Exploder).