As a result, I currently use languages with decent type systems - Rust, recently F# - but I've still never come across a reason to know that something is a monoid, and I don't regularly come across duplicated code, so I'm not exactly sure what problem knowing something is a monoid is supposed to solve in my domain.
The CRDTs linked by the parent are a great example. If you're working on an eventually consistent system, and you see a structure that doesn't form a monoid, it's most likely the case that there's going to be some sort of race condition. That isn't to say that any given monoid is going to solve the problem, but it can definitely help to pinpoint possible problems in otherwise complicated code.
It definitely doesn't come up in every discipline, but studying these structures has improved my engineering a ton. If design patterns are about class and object composition, then I'd argue that algebraic structures are the equivalent for function composition.
Full disclosure: I write mostly functional Erlang code for a living
(It is unfortunate that monoids sound like monad's more complicated brother, when in fact it is exactly the opposite way around. Monoids are so simple that a common reaction once you actually get it is to go "Seriously? That's it?")
The other useful thing about monoids is that because of the associativity, they provide a useful, yet easy, way of breaking up a task for multiprocessing. If you can prove that your data type and the operations you desire to use on it are monoidal (and by "prove" I mean the informal sort of things programmers do all the time, not grabbing a proof system and going at it), then you have some really easy options, and it's really easy to write a system that very cleanly breaks up "the strategy of how I'm going to multiprocess this" from "what it is I want to multiprocess".
Because of this, it is likely that you're going to continue hearing more about these things over time, rather than less. Monoids are just about the simplest non-trivial example of such things, and the more sophisticated steps up (lattices, CRDTs [1]) are probably also things you're going to hear more about in the future.
Personally, I think in a lot of ways we are still have collectively done next to nothing as a community to address the multicore world we live in. Right now we're taking advantage of some things like Go's green threads or "asynchronous" programming to deal with the common case where we have some problem where a single core is adequate to solve it, but we need some complicated scheduling or we have huge numbers of these relatively small problems coming at us at once, but we still have made very little progress in doing complex tasks truly in parallel. It is very likely that those solutions are going to involve coming to understand these mathematical abstractions and why they are important. Monoid is going to be the first of them you'll encounter, because it is, as I said, probably the simplest non-trivial one. But on its own it won't get you all that far in general.
[1]: https://en.wikipedia.org/wiki/Conflict-free_replicated_data_...