Let $M$ denote a monoid. Then to refer to submonoids of $M$ that just happens to be a group, I think the phrase "subgroup of $M$" is okay, as it's unlikely to cause confusion as long as you instruct the reader you'll be using the word in this way.
However, sometimes you've got a monoid $M$ with a subsemigroup $M$ that just happens to be a group, but whose identity element is different to that of $M$. This happens with sandpiles for example; there's a monoid $M$ of sandpiles, and this has a special "subgroup", but the subgroup has a different identity element to $M$.
Question. Is there a term for this?
In semigroup theory, the accepted terminology is the following. A subgroup of a semigroup $S$ is a subsemigroup of $S$ that happens to be a group. In particular, its identity is an idempotent, although $S$ might have no identity.
In $S$ is a monoid with identity $1$, a subgroup whose identity is $1$ is a subgroup of the group of units of $S$. You may also simply call it a subgroup with identity $1$.
That being said, I agree that this leads to some confusion. For instance, how do you define a submonoid of a monoid? The official definition insists on having the same identity. Thus you arrive to the conclusion that a subgroup of a monoid $M$ is not necessarily a submonoid of $M$. Furthermore, if $S$ is a semigroup and $e$ is an idempotent, the subsemigroup $eSe$ happens to be a monoid with identity $e$, sometimes called the local submonoid of $S$ at $e$.