Yes it is entirely possible to poop out of your mouth and it's called fecal vomiting and is an indicator of a serious health issue that needs medical attention.
If you start noticing that your breath smells of poop then it's likely you're gonna be suffering a serious health issue soon so you should let your doctor know about it if you notice your breath smelling like poop.
If you do poop out of your mouth as in vomiting poop then you should go to urgent care or see your doctor soon about it because it's serious and should not be happening.
The poop won't come out like it will your anus though but in the form of vomit.
The fecal vomiting will also have poop on the breath smell and other gastrointestinal symptoms, including abdominal pain, abdominal distension, dehydration, and diarrhea.
It's disgusting and thankfully it's never happened to me or anyone in my family but it happened to a friend in school years ago and the school called 911 and took her to the hospital.
She was okay later though.