I graduated in May 2019 (full-time student) and then started working in July. For the 2019 taxes, should I enter my occupation as a student or my current profession?

Makes no difference.  You'll still have a W-2 to enter from your profession (job) and student deductions.  So you can put either as your profession.

I recommend filing your occupation as of December 31, 2019.  It is used for statistical purposes and does not impact your taxes.