Black–Scholes is the heat equation backwards, which has pretty different behaviour to the heat equation as the latter smooths things out over time and the former makes them less smooth over time. But this does make some sense: when an option reaches expiry you know exactly how much it’s worth (as a function of strike price) but the further you are before, the less well you can predict the strike price and the smoother the price function should be. Indeed your substitution reverses the direction of time but intuitions about the heat equation aren’t so applicable to Black–Scholes because intuitions are often directional.
The difference is that in the Schrödinger case you're effectively 'turning' the solution (in the complex plane) which leads to the uncomfortable question of whether the solution to the heat equation you'd start with is still defined. When going from heat to Black-Scholes you're just rescaling in 'existing' dimensions which doesn't change the character of the PDE.