In most cases, designations like physicist, mathematician, chemist, and biologist are reserved for those doing active research in their field. Most of the time, a mathematician would have a record of scholarly papers that resulted from their research - though other evidences of active scholarship might also suffice - patents, inventions, etc. Since most PhDs require significant research, a PhD in mathematics would also tend to qualify one to describe themselves as a mathematician.
Even though I served on the Math faculty of the Air Force Academy, I never really considered myself a mathematician, but rather as a mathematical scientist - one who used existing mathematics to address numerous interesting problems in various natural sciences - physics, brain injury, ballistics, and fisheries science mostly, but also taking forays into environmental sciences and other areas. For me, a mathematician is one who develops new mathematics.
One student I mentor works as a math tutor and has published seven peer-reviewed papers spanning several areas of physics and mechanical engineering. Given his scholarly record, it would be appropriate for him to refer to himself as either a physicist or a mechanical engineer. Referring to himself as a mathematician would not be quite right.
Another student I've mentored has published most of her scholarly work in either chemistry or physics. She has one published paper in mathematics, but given the lack of ongoing work in mathematics and her current research focus in chemistry, she tends to view herself as a chemist.
At least in the US, scholarship and publication (rather than teaching) tend to be the key things people expect when one has a designation as a mathematician or specific kind of scientist.
