"La fille se met droit" (X) doesn't mean the same thing as "La fille se met debout" because the meaning of the former is vague/unclear and the phrase/sentence is not idiomatic or incomplete, as you know, whereas the latter means she gets up and is idiomatic and complete.
Ac. and Robert are defining/explaining the meaning of the verb (se) dresser, and not the other way around. "Se mettre droit" evokes the motion or positioning towards (greater) verticality and helps to understand those contexts where "se dresser" is used in that entry. The only way I can imagine a human being "se mettre droit" per se is when they're curled up in bed and a doctor asks them to stretch their legs and be on their back, but they would ask "allongez vos jambes et mettez-vous sur le dos" and not "mettez-vous droit", and in any case it wouldn't be equivalent to "se dresser" in their bed.
You can find on the Web a few instances where "mets-toi droit" is used for "tiens-toi droit" and this may be the result of a poor/automatic translation, lack of proficiency or very relaxed speech with dropping of a complement; maybe some speakers feel the need to resort to mettre instead of tenir when the person is not standing up. There may even be a regional component to this. Otherwise inanimate objects can possibly come back to their non-folded/upright/center/in-line position and "se mettre droit" could possibly describe this. Something like "se mettre droit devant la caméra" would be in front, lined up with, not standing tall in front of.
Incidentally, there is no connection with the "to put straight" idiom in English.