Yes, its so often that we see batman acting like royalty and like everyone is inferior and should be serving him. Thats Bruce in a nutshell to you?
Id argue it wasnt an inversion of roles, more like a big brother little brother dynamic that had its own appeal.that's why the Damian-Dick partnership worked so well as it was the inversion of the Batman-Robin dynamic.
A trait they share with the entirety of the hero community.As for Damian and Bruce they are both over-dramatic
Who isnt?stubborn
Actually Damian is a Tsundere, Bruce is just clumsy at social interactions.dominant personalities who are not expressive with their feelings
Who doesnt?but feel very deeply nonetheless
Like any other batfam member, and no more than jason actually.They place great importance on family
Wouldnt say its compulsory, tho he does want to take care of his family. Also nice comparing Robins to pets, really apt comparison there. To Bruce they are like sons, to Damian they are like toys.(e.g., Bruce has a compulsory of adopting children which is mirrored by Damian's tendency to hord pets)
So nothing like Bruce.And lastly this is how Morrsion describes him:"[Damian Wayne] has been raised by monsters and fiends
Note that he wants to be the son of batman, he doesnt want to become Bruce. And even if he did, he is still far too different for this to ever be a thing.but inside he’s really a good person because he’s the son of Batman and that’s who he really wants to be."