most versions say "The Banks of Sweet Dundee," not "THE sweet Dundee," so we don't have to assume that Dundee is the name of the river.

Since when do towns have 'banks' (apart from the financial sort, of course)???

No, the name is of a river. Unforunately it doesn't exist... The truth is nobody knows.

I think we're best quoting the admirable Steve Roud:
(copied from http://www.tradsong.org/peri.htm)

Ruairidh Greig, suggested that the former should actually be spelt The Banks of the Sweet Dun Dee, because the River Dee was once a dirty brown colour, but I was not convinced by this suggestion however ingenious it may sound, and I cannot find any evidence that the Dee was ever that colour. But his suggestion proved to be just the clue that I needed. On the way home I mused over the problem and with the aid of my indexes (of which you may have heard) and my expert knowledge of the subject of Folk Song, I can clear up this mystery. In fact, the 'Dee' part of the tide is a red herring as it was introduced into the song by a short-sighted broadside printer in the early 19th cent". This can be easily tested. Working by candlelight, take your glasses off, close your good eye, and read the title of the Catnach broadside of the song and you will discover that it should read Banks of the Sweet Dun Cow - and this is obviously what it was originally. This makes the whole thing quite clear, and indeed it sheds important light on the real meaning (hidden for over 100 years) of the whole song. By checking the standard social histories of the 19th century, one can easily discover that when the Rochdale Pioneers formed the Co-op Bank they couldn't afford proper premises and anyway all the decent corner properties were already owned by their rivals, so they hired a room in the back of the pub called the Dun Cow, and their potential customers were thus exhorted to 'bank at the sweet Dun Cow' (I am not sure why it was called the sweet Dun Cow - perhaps there were two in the town and this one also sold confectionery). In the light of this new knowledge one can see that the whole song is about the working classes (given the heroic title Undaunted Mary) slaying the capitalist bosses (the Squire) and thereby gaining their wealth.