I have recently been taught the Fokker-Planck equation for diffusion, which says, for $t \geq s$: $$ \frac{\partial}{\partial t} p(x; t|y; s) = \frac{\partial}{\partial x} (f(x, t)p(x; t|y; s)) + \frac{1}{2}\frac{\partial^2}{\partial x^2}(g^2(x, t)p(x; t|y; s))$$
where $f(x, t)$ and $g(x, t)$ are the drift and the diffusion coefficients, respectively. We derived this equation by taking the limit of a discrete Markov process. However, I notice that I can keep the right hand side of the equation the same, but change the drift and the diffusion coefficients.
For example I can write: $$\frac{\partial}{\partial t} p(x; t|y; s) = \frac{\partial}{\partial x} (f'(x, t)p(x; t|y; s))$$ where $f'(x, t) = f(x, t) + \frac{1}{2}\frac{\frac{\partial}{\partial x} (g^2(x, t)p(x; t|y; s))}{p(x; t|y; s)},$ so that it has a zero diffusion coefficient and a different drift. We can obviously also rearrange for it to have a non-zero diffusion coefficient.
It seems we can have multiple drift and diffusion pairs such that the evolution of the conditional probability density function is the same. But intuitively it seems that given the same conditional density function evolution, the diffusion process should also be the same.
I asked my teacher about it, and they said that we just consider the "simplest" form, but I don't find it satisfactory. I think I'm missing something very obvious here. Could someone please help me?