I believe in some fields like physics or other experimental science, hiring a student without PhD degree as a research assistant is common. In pure mathematics, do professors have the need to hire non-PhD students as their research assistants?
In pure Mathematics, there is less need for research assistants. The typical jobs of an RA such as conducting experiments, wrangling data sets, doing literary search, are not there. Research in pure Mathematics tends to be very specialized. In some research areas like algebraic number theory or algebraic geometry, you can spend years just to get to the current margin of research. A professor in those areas needs to groom students in order to work with them. There are some exceptions, such as graph theory or combinatorics, where it is in general easier to understand a research question. I got my Ph.D. in non-associative algebra and I would never have been able to use a non-Ph.D. research assistant. Maybe if I would have been interested in working in computer algebra, I could have used a non-Ph.D. research assistant.
Also, funding in pure Mathematics is very restricted. NSF grants in pure Mathematics tend to not contain funding for research assistants that are NOT Ph.D. candidates, but this is the case for other disciplines as well. Undergraduate research is a different subject. A professor in pure Mathematics will have to think of something that undergraduates actually can do, and it would typically not be in their main line of work.
TLTR: My experience suggests that there is no need for research assistants in pure Mathematics.
In pure mathematics, student employees don't usually "help" their advisors do the adviser's research. They generally do their own but on a topic of interest to the advisor, perhaps extending their work. Applied math is potentially different.
But RA isn't a globally defined term. A university, even an undergraduate university could use it as a title for an arrangement in which the student does some research and gets some stipend. Thus it is a term for a certain kind of funding that has few external duties beyond research.
So, in that sense, a student other than a PhD student might get such an award. But not in every institution. And, the funding might be entirely internal to a department. Perhaps a really good student has a really severe financial constraint (poor as dirt). A department might be able to do some things while giving a student opportunities. Rare, but possible. And it would probably be the department that "hired" the student (gave the grant), not a professor.