I know exactly what a septic tank is, I just replaced the one at my Dad's house. How does that term relate to a nickname for an American?
I'm familiar with the origin of "Digger", "Doughboy", "Yank", and "Limey" but I've never heard of the term "Septic" used as a cultural nickname.
I think it is a rhyme. If you speak "septic tank" fast enough such as in Australian English, it is a rhyme with Yank. This is why Australians started to use it as a codename for Yanks, and then dropping the second word "tank" altogether.
It is very distinctively Australian English - one which even we across the Tasman don't use.