Many cities and counties require homeowners with swimming pools to have fences of a certain height, etc. to discourge folks, particularly kids, from just "jumping in" and possibly drowning, while the homeowner is away. It may be that the city or county told him his existing fence, even if in good condition, did not conform to their specifications. It also could have been his homeowner's insurance. Or he may simply have liked this style of fence better. There is no way to know with absolute certainty what his motivation was. Just be aware that perhaps it was not as "optional" as it looks.
I don't think the fact this is a "boundary" fence would necessarily give him the legal right to prohibit her from leaning plywood against it. However, if she did, and the plywood damaged the fence, he likely could sue her for damages. It probably would wind up in small claims court.
Obviously, we don't know how much plywood your mother has or why she prefers to lean it against the fence versus doing something else with it. But, if it were up to me, I would honor the neighbor's request.