Monads have no inherent connection to any notion of "the world"; that's just a thing for Haskell's IO monad specifically (and perhaps derivatives thereof) because what that abstracts over is effects that are external to the program.

A monad is any construct of abstraction that obeys the Monad Laws. This may seem a useless definition, but my point is that deriving a useful intuition for the concept of the monad is necessarily challenging because they cannot readily be boiled down any further than that without loss of information.

The best intuitive explanation I've developed so far is that a monad is an abstraction over structures of ordered computations. The most ready examples for non-functional programmers are probably exceptions (and their handlers) and nullable types (typically called "optional types" in languages with first-class handling of such things, unlike Java).

Not sure if going back to C++ is the answer, given how much Haskell like code one can write since C++20, and it has gotten even better with newer standards.