I work at a company that makes sites for different credit unions / banks. The way we solved this was by identifying each Financial Institution (FI) first by the url and then displaying all related data for that fi.
sampleurl.com/default.aspx?fi=FI1
sampleurl.com/default.aspx?fi=FI2
For the custom theme portion, I created a dynamic CSS stylesheet that contains all of the color information for that fi, like link color, background color images and so on.... All of which is added at runtime. Granted, our approach is using WebForms once having the idea of how things should be laid out converting them to MVC shouldn't be too hard.
Here are some samples:
Obee Credit Union
Our Demo Site