Accreditation is most likely not the reason for job postings of this sort. The Higher Learning Commission and most State Bodies (such as Ohio Board of Regents) give universities and colleges a great deal of latitude regarding whom they hire and tenure. It's only the professional programs involving licenses that will usually nitpick about education background: Teacher Education, Counseling Education, Athletic Training, School Psychology, etc. etc.
From personal experience in hiring faculty members, I would say that the most likely reason why these job postings are specifying Ph.D.'s in Mathematics is that in those situations they would have a fairly good idea of what course of study the given candidate would have taken. A Mathematics Department usually wants someone who can teach a variety of courses from the rudimentary to the specialized, and a Ph.D. in Mathematics would guarantee that the candidate personally experienced this range as a student.
Furthermore, a Mathematics Department usually wants someone with a specific mathematical research agenda, optimally one that complements their assigned teaching well. (Actually, the interface of Teaching and Research for Faculty Positions will of course differ greatly from school to school) The easiest way to guarantee this is to hire someone with the most obvious educational background.
That all said, I would encourage your colleague to apply for any position that they think they would be a good fit for, whether the posting specifies a math degree or not. The worst that can happen is that they hire someone else.