Let's say the coupling of $\mu$ and $\nu$ gives almost all the mass to the diagonal with just a little bump under it. They will satisfy the stochastic ordering, but how their neighborhoods will happen to satisfy it as well?

no, it's not. since there will be some measures in a neighborhood of $\mu$ that are not dominated by $\mu$. My point is that your definition per se only seems to work for discrete topology regardless of the order, since in particular it must work for $\nu = \mu$

isn't it the case that that's the only topology valid in this case then?

what if $\mu = \nu$?

I am not an expert on continuous-time Markov processes, but I guess in discrete time if I have observations of $Y$ (being constructed dynamically using $Y$ and $X$), it is not enough for me to just know the latest value of $Y_t$ to get everything I can for the distribution of $X_t$, in most cases knowing the whole trajectory of $Y$ provides a strictly finer conditioning. I guess the same woule apply to continous-time case. Namely, $\Bbb P(X_t|\mathcal F_t^Y)$ is not a function of simply $Y_t$ and $t$.

I took a break from academia a year after asking this question, so libraries are not of great availability to me. Thanks nevertheless