Each node has 4 edges, so is visited twice. We start and end on node A, but it must be visited some time in between as well.
Let's split this up into cases depending on how many other nodes are visited before we come back to A the first time.
This gives a total of:
More explicitly, the routes are:
Edit : This is an answer to the question:
How many different paths exist from A to A that go through each other point exactly once ?
However, per the comments, it seems the OP is looking for something else. I cannot figure out what exactly yet.
paths that qualify.
Okay. THAT. WAS. INCREDIBLE!
It took me two solid hours of work to solve and even longer to write and draw it all up here - hopefully it'll be worth it! To start us off, here is the final maze layout and routes:
In the following explanation all colours have been abbreviated to their initials as follows: G=Green, O=Orange, P=Purple, Y=Yellow.