Having studied abstract algebra before using FP, I must say that I instantly made this connection when I came into contact with monads, and that it helped me a lot to understand them.
So I wouldn't be so quick to dismiss the idea of associating these concepts. After all, category theory is all about making that kind of associations.
This is a good observation. Though, most programmers don't study, and don't have to study, enough abstract algebra before being exposed to promises, optionals, and even lists (which all exhibit some form of monadic behavior).
What worked best for me was both playing with code, reading the definitions in code, and then look up some CT to get to the mathematical roots and where they fit in the intuitive picture. (Then you can continue with arrows, monad transformers, etc.)
Since my primary purpose for learning monads (and other algebraic constructs) was and is their use in programming, I concentrate on "practical" intuitions, and periodically check if I can go down to the algebraic base of them in a particular case. If I still can, I suppose I know enough math to get by in the particular area. If not, I open a book and clarify my understanding, and maybe glean something new.
So I wouldn't be so quick to dismiss the idea of associating these concepts. After all, category theory is all about making that kind of associations.