I’ve never seen “professional mother” before. My guess is that the phrase might be used by a woman who stays home to raise the kids, but doesn’t want to be seen as somehow less important than a woman who has a paid job. So, the woman might — either seriously or maybe jokingly — call herself a “professional mother” as a way of saying “being a full-time mom is just as important a job as any paid job.”

There's two options, so context will help to figure out which is correct: 1. The woman is a working mother. She has a full-time (that is, 40hrs/week) career in addition to her duties as a caregiver to her children. This is considered quite impressive if done well! It's already a lot of work to raise a child, and working full-time as can be very stressful. 2. The woman is a "full-time" mother or "stay-at-home" mom. In other words, her only duty is to raise her children. Here, "professional mother" essentially means that mothering is her profession. Many women choose to do this, although the practice is becoming less common in modern times. Is it common? Not incredibly. I would expect someone to use one of the alternate versions I used at the beginning of each paragraph.