How about just using the callback of one to deselect the other:
% Set up some example listboxes:
u = uifigure;
lsta = uilistbox(u,'Items',{'Cat' 'Dog'});
lstb = uilistbox(u,'Items',{'Monkey' 'Horse'});
lsta.Position(2) = sum(lstb.Position([2 4]));
% initial state
lstb.Value = {};
% Changing value of lsta deselects lstb (and vice versa)
lsta.ValueChangedFcn = @(~,~)set(lstb,'Value',{});
lstb.ValueChangedFcn = @(~,~)set(lsta,'Value',{});
