In the US at least, it is entirely possible to teach at a university without a PhD. Community colleges are full of instructors with masters's degrees, and tons of classes offered by major universities are taught by graduate students or adjunct faculty without doctorates.
Your job title probably won't be 'professor', but you'll be doing basically the same work as one.
Graduate students teach classes at their own universities as part of their departmental funding. This is only a temporary situation and exists only while they're enrolled. It's not a career path.
As a former graduate student myself, I'm actually not aware of any non-PhDs who are adjuct faculty or community college instructors. I'm not claiming that they don't exist anywhere, but given the number of PhDs and the number of available academic jobs, the competition is fierce, and non-PhD candidates are likely to lose out to PhD candidates.
Fwiw my dad had a masters in biology and a PhD in botany, but was an instructor for biology in the local community college (“Mount San Jacinto Community College”). I guess technically he had a PhD, but not in the way most people would think
Your job title probably won't be 'professor', but you'll be doing basically the same work as one.