I do actually call my mom bro, but if like, a friend or coworker said "don't call me that" I just wouldn't do it.

Edit: this is tangential to the real point of the post, which is just to not call people things they don't like.

Bro is harder to argue for sure.

And man, unless it's more, "oh man, that's rough" as an excalamatory rather than "good to see you man" is still gendered.

But dude has never been gendered. It was mostly used by guys towards guys, but the origins of that usage (rather than dude ranches or the derogatory term related to that) it was applied to everyone. Dudette came along later but was essentially created because the usage was male dominated, not because dude was gendered. It's one of the rare gender neutral, inclusive slang terms. So much so that when dudette was thrown around, it got rejected as unnecessary, and was sometimes taken offensively. Same with dudina and dudess.

Mind you, the era where it was mostly an underground slang used in African American circles is murkier, since it was underground, less written at the time, and after it got "borrowed" by white kids lost its popularity there.

But when surfer culture picked it up, and it spread via movies, female surfers were called dude, and used it the same way as female surfers. They were just such a minority that the association didn't stick in pop culture because what got seen was Spicoli, and the association with it as being used by guys about guys got absorbed as the primary usage.

There was no gender division in that origin, nor was there a need for it. There simply wasn't a female specific alternative to dude.

Since it is still used inclusively far more than it isn't, it's usually better to assume the best rather than the worst. Someone duding someone in a casual and friendly way is unlikely to be using it as a gendered term. It's more like buddy, or pal, or even mate than something like bro that started gendered and is still predominantly used that way.