This was intended to be a comment, but is too long so I will post it as an answer. First of all, a disclaimer, I am a physict and all that I know about quantitative finance comes from self-learning, so please feel free of correcting me if I am mistaken (also in the physics stuff, of course!).
I have been doing a little of research, and perhaps you are right in your last point. The Black-Scholes PDE relies in the assumption of (i) the option prize is a continuous function of time and the undelying asset and (ii) the current stock price follows a [geometric] Brownian motion. From the physics point of view, there is a deep connection between Brownian motion and the diffusion equation which is exemplified in the famous Einstein relation. As the heat equation is a particular form of the diffusion equation, it is not so surprising that the heat kernel appears in some of the solutions of the Black-Scholes PDE.
However, there is no physical similarity between these two phenomena as they both do not describe physical processes but only are based in the same mathematics [stochastic calculus].
As a curious fact, I was reading not long time ago the book When Genius Failed: The Rise and Fall of Long-Term Capital Management. There it is said that Myron Scholes and specially Robert C. Merton developped all their option pricing theory mimicking physical models [taken from research papers and books of statistical mechanics] and having faith in the so-called "efficient market hypothesis". The history of LTCM is well-known in finance, they lose billions of dollars after having been leveraged $250$ to $1$ [that is they invested $250$ dollars while having actually $1$].