They are both regression methods for discrete dependent variables.
I have a discrete dependent variables like $\{A, B, C, D, E\}$, and $A$, $B$, $C$, $D$, $E$ can be cardinal data(10, 20, 30...) or non-cardinal data(like car, train, and ship).
Which model is proper for each case?
Multinomial regression suits nominal outcome data like your second example (car, train, ship...tricycle...), but ignores any order in those categories. I suppose it could work for any set of discrete outcomes, but it would waste information and estimate model parameters inefficiently if used to model ordinal (ranked) or cardinal (count) data. Ordinal logistic regression is better for ordered categorical outcome data.
Poisson regression is probably better for modeling cardinal data, but it isn't optimal for all count data. The Poisson distribution's expected value is equal to its variance, which can be a problem (e.g., with dependent observations or an excess of zeroes). Negative binomial regression can accommodate overdispersed counts better (see also "Poisson or quasi poisson in a regression with count data and overdispersion?"). Zero-inflated models help handle datasets with lots of counts equal to zero, though this requires more statistical power to do well.
