For Cpt. America this makes total sense...his shield is both a weapon and a defensive device, as demonstrated throughout the movies many times.

But carrying a small dog (a pug) while charging into battle? What exactly is the point? Where is the logic? Even if its the royal dog, when the compound is overrun by the archenemy of the duke, the royal family unaccounted for, the most effective weapons sabotaged, the last thing his most capable military commanders would think of doing, is saving the royal pet. And even if that was the intention, how does carrying it into the thick of the fight achieve that?

Yes, we accept absudities in movies, if they make sense in the setting. If Cpt. Rogers were to carry a pet hamster around in his pocket for seemingly no reason, while fighting Hydra, it bet the acceptance of this would be kinda low, Marvel or no.

Any story has an implicit assumption about the kinds and levels of absurdity you can expect. People get weirded out when you violate those expectations. Marvel, by virtue of their source material if nothing else, has set a much higher baseline of goofiness for their movies than most others. Whereas we all sort of know Dune was meant to be Serious Scifi. And I guess people who decide they like the Lynch adaptation have re-calibrated their expectations, and found the movie internally consistent on some level. It's all about audience expectations (and sometimes Hollywood ignorance/laziness, I guess).