Would it help if you defined a monoid as a combination of 3 things?

1) a data type A

2) an associative operation on A

3) an identity (or empty element)

Then you can correctly say that the string data type, admits an associative operation (concatenation of two strings) and you have an empty element (the empty string).

I think too many people talking about functional programming really overblow how much you need to understand about mathematics, they serious do.

Think about my definition and you can quickly understand that there are many monoid instances for numbers (with the associative operation being addition to get a monoid or multiplication to get another monoid instance).

There's infinite numbers of monoid instances for various data types.

Haskell has several shortcomings from what I remember and Haskell developers incorrectly assume that you can only have one semi group (or equality, monoid, etc) instances for your data type because they believe they are the closest to math, but in reality their language has its own limitations.