For those who are interested in the field - whether as art, for the fun of folding or the various potential practical applications - the best book on the subject by far is Robert Lang’s Origami Design Secrets, which is a fantastic read and covers all of these and more. Lang is a mathematician at heart but also a consummate artist. I certainly can’t recommend origami enough as a hobby.
More info on Robert Harbin: 
I'd love to know if people blended origami, pepakura and mechanics
It takes quite a lot of time to watch (it's a full semester course) but definitely worth your time if you're into origami and mathematics.
It makes me wonder if a formal naming process is also possible, given that origami is a largely linear, and well defined process.
You'd start with a prefix which denotes the shape of the paper (square or aspect ratio). Then lists the required transforms in order. There would be some ambiguity as order of certain transformations might not be critical. Perhaps you could develop standard notation there.
I see that there appears to be some academic software for modeling origami structures. So I assume this problem must have been addressed to some extent:
Robby Kraft's Rabbit Ear software is very nice: https://rabbitear.org/
Not saying it's intractable, just more nuanced than it first appears.