Puberty can start before you can see it. My oldest boy and girl started puberty really early and one of the first things we noticed was smell. Maybe at this age deodorant is more sensible than anti-perspirant and there are more natural products than sprays (and more eco friendly) if that would be better. Regardless of whether you think it is necessary or not, I think Mum is being very sensible because their friends would certainly notice if they smell different and they could easily become a target for teasing. If she can make showering and deodorising a habit at this age she won't have the arguments later like a lot of parents do.
Last weekend, in Rutland, the first statue in Britain of the late Elizabeth II was unveiled.