If I take everyone's meanings correctly, what you describe (a monoid with the identity removed) is different from what cottonseed (https://news.ycombinator.com/item?id=12110936) describes, which is a monoid with the identity forgotten. That is, it's still there, but we don't require it to be there axiomatically.
Note that every monoid is a semigroup (because it satisfies a stronger collection of axioms), but not every monoid with identity removed is a semigroup (because it's possible to combine two non-identity elements to get the identity element): consider, for example, the integers without 0.
If I take everyone's meanings correctly, what you describe (a monoid with the identity removed) is different from what cottonseed (https://news.ycombinator.com/item?id=12110936) describes, which is a monoid with the identity forgotten. That is, it's still there, but we don't require it to be there axiomatically.
Note that every monoid is a semigroup (because it satisfies a stronger collection of axioms), but not every monoid with identity removed is a semigroup (because it's possible to combine two non-identity elements to get the identity element): consider, for example, the integers without 0.