As literary forms, these three modes of writing are rather distinct from one another. The similarities shared between them are general and can be extended to include long form fiction as well. Short stories, drama, poetry and novels are each examples of creative literary forms.
Each of these forms is a mode of creative writing. Imagination is at the heart of each genre. Many literary tools are shared between the short story, drama and poetry such as a use of metaphor and other types of figurative language.
Drama and shorts stories can be argued to, formally, have more in common with one another than they do with poetry because drama and short stories are narrative - they tell a story, feature characters (and characterization) and present both conflict and resolution.
Poetry can be narrative but is not defined by the presence of a narrative as these other modes are. There is no such thing as a non-narrative short story because narrative is essential to the definition of a short story. The same thing goes for drama. However, poetry is definitively still poetry regardless of the presence of narrative (character, conflict, resolution, etc.).
They are the same, they all tell us something imaginary or real and they involve characters and problems. The only diffrence is how they tell it. Drama is all about dialogue and acting, poetry is about rhyme and style, and short story is about being simple, quick and still being able to tell a message.