For those who might read this in the future…
After Ven’s comment, I take a look into how multivariate modelling works in brms. I tried to apply the syntax that was introduced here. The model runs but the issue is that for each equation the parameters are estimated separately. For example, in my case, I got replicated ASCs:
choicebus_asc1_Intercept
choicerail_asc1_Intercept
Then I tried to explore options to get around this issue and make some parameters shared where I saw this post where someone was trying to implement shared parameters in non-linear multivariate equations. But sadly, it is not possible in the current version.
However, the feature is expected in brms 3.0. Hopefully, I can revisit this quest then :)