Most words have multiple meanings depending on context; professional is no different.
(AHD) professional, adj.
a. Of, relating to, engaged in, or suitable for a profession: lawyers, doctors, and other professional people.
b. Conforming to the standards of a profession: professional behavior.
2. Engaging in a given activity as a source of livelihood or as a career: a professional writer.
3. Performed by persons receiving pay: professional football.
4. Having or showing great skill; expert: a professional repair job.
Profession, according to Etymonline.com, originated in one's declaration to join a Catholic religious institute:
c.1200, "vows taken upon entering a religious order," from Old French profession (12c.), from Latin professionem (nominative professio) "public declaration," from past participle stem of profiteri "declare openly" (see profess). Meaning "any solemn declaration" is from mid-14c. Meaning "occupation one professes to be skilled in" is from early 15c.; meaning "body of persons engaged in some occupation" is from 1610; as a euphemism for "prostitution" (compare oldest profession) it is recorded from 1888.
For the adjective form professional,
early 15c., of religious orders; 1747 of careers (especially of the skilled or learned trades from c.1793); see profession.
Professor, referring to a university instructor of a certain rank, is related, but is borrowed via Old French directly from Latin:
person who professes to be an expert in some art or science; teacher of highest rank, the agent noun from profiteri, "lay claim to, declare openly" (see profess). As a title prefixed to a name, it dates from 1706.
So it seems the sense of professional as one in an occupation requiring special training, skills, or education is the original one, and it is only loosely related to professor.
And as with most words, context is important in how you would use professional.
In sports parlance, for example, a professional is one who is paid to play a sport as opposed to an amateur who trains as a pastime. This notion is independent of any skill level or training, as evidenced by the controversy over the treatment of student-athletes in the NCAA, and by the New York Mets. Between the two is the world of semi-professional sports, where competitors are paid, but do not train and play as a full-time profession, and must take at least part-time employment to meet their expenses.
In the military, in contrast, a professional army is comprised of enlistees who make the military their full-time career, as opposed to private citizens who volunteer or are conscripted (a citizen army).
In economics, political science, and other social sciences, professional refers to a class of workers with advanced education and specialized training: medicine, law, engineering, academia, and so on. By this definition, an administrative assistant would not be a professional even though he is paid to do the job, and may wear a tie and work in an office; the job requires no specialized degree or advanced training.
In still broader contexts, professional may be synonymous with white-collar worker, including not just doctors and lawyers but people like consultants, analysts, or really anyone who works at a desk. When economic development boards and fashionable apartment buildings promote their friendliness to single "young professionals" it is in this sense of the word.