I am not sure why you would think that teaching is not a profession, I would say that it is one of the oldest vocations around.
Signs of education and learning can be seen in all of the ancient civilisations.
Plato, the ancient Greek philosopher founded his own academy, "The Lyceum" in Athens in 387 BC. These early schools set the president for future learning, and the importance of teaching and academia.
You might define a profession as different from just a "job" as it requires training and often a recognised qualification.
By this definition, teaching could certainly be described as a profession as teachers are required to have completed standard schooling to the age of 18, followed by a university degree, either in Education itself, or in another subject, followed by a years Post Graduate study which qualifies them as a teacher.