A "mathematical object" in this case is a description of how the object is expected to behave. But even so, I think something is lost in reducing monads to a c

A "mathematical object" in this case is a description of how the object is expected to behave. But even so, I think something is lost in reducing monads to a clump of related behaviors and properties. You still need intuition about what these behaviors actually mean and what kinds of real-world objects actually conform to these properties. So you are right that monad "is not" a nondeterministic computation or a container or whatever, but a monad "is" a common set of behaviors/properties that we should expect nondeterministic computations and containers to conform to. Abstraction without intuition is obfuscation!