It's clearly more strongly a monoid. Is this a historical thing or something abstract I'm missing?
Using the term "semigroup" does not commit us to considering identity as a part of it. Sure, we give it notation $T_0$ or $S(0)$ or such when it's convenient notationally, but next minute we may want to exclude it from consideration. For example, one says "semigroup of compact operators on [a Banach space]" while knowing perfectly well that the identity isn't compact; it's just being ignored in this statement. Saying "monoid of compact operators", which explicitly includes identity, would be quite strange.
When I consider the Poisson semigroup on the circle (that's when you extend the function harmonically to the disk, and then restrict to a smaller concentric circle), it somehow doesn't occur to me that I should include identity as a part of semigroup, let alone to emphasize its membership by saying "monoid". That's because the Poisson semigroup is about convolution with the Poisson kernel, and the identity operator isn't. While the behavior as $t\to 0^+$ is important, including identity as a member of the semigroup does not help me the slightest bit in studying that behavior. (This goes for the heat semigroup, too.)
Summary: It makes sense to use terms that emphasize the structure that matters. The membership status of identity in operator semigroups does not matter from the analytic point of view. It typically does not share the properties of other operators in the semigroup.