I will make a guess. If the payoffs are level invariant as in your examples, and you have 'continuous monitoring', and no cost-of-carry parameters, then what you want might follow simply from dimensional analysis of the value function. After all, you need dimensionless factors. But once you introduce discrete-time monitoring at spacings [$]\Delta t[$], then potentially dimensionless factors like [$]\sigma^2 \Delta t[$] and [$]\sigma^2 T[$] could both appear, destroying the argument. Like I said -- a guess.