A dog is not an accessory.
Period.
First, I feel it necessary to say I am a dog owner, and I have a nurturing love-hate relationship with my dogs. I love them but hate the stupidity and behavioral problems they sometimes have. And before anyone points a singular stinky digit at me with the usual "they are only as bad as you let them" or "you never trained them right," listen: I read the goddamn puppy books--Your Puppy's First Year and so on and so on, and I have been consistent and strict with them. I have two dogs who are sisters. One is smart, highly independent, and moody; the other is super-dumb and eager to please with a crazy wild streak. (If she were a child, she'd be the helmeted kid tethered to his mother at age 13.) So the constant challenge is trying to simultaneously train and discipline two dogs that are polar opposites.
But I am getting off the point. A dog is not an accessory.
The last time dogs were an actual accessory was when the French aristocracy had little lap dogs around to lure the fleas away from themselves. (Yes, it's true.)
You shouldn't carry a pet around all the time.
It's not cute. You either have what is basically a meshy purse with a claustrophobic dog peeking out, or you have a person with a handful of dog ass or crotch.
You've heard all of the studies about people not washing their hands after going to the bathroom. Well, add in all the people carting around their little chihuahuas, daschunds, pomeranians, and other nervous-pissing little gasbags touching all the stuff you want, the stuff you don't want anymore, the doorhandles on the exits you so desperately want to use.
Dogs don't wear diapers, and they shouldn't. And they definitely shouldn't be brought places where they could spontaneously need one.
Some dogs get cramped up, they have literal shit fits or pee themselves if scared... and they might bite someone if they really dislike them. Seeing-eye dogs aside (which are strictly and specially trained), one shouldn't bring pets into the public domain unnecessarily. You don't run into people like that in public. You know why? we put people like that into asylums or home care lockdown.
Let's mention something else here: Allergies.
It is fucking rude to carry around what is a severe allergen for some
people in public places. Put on the imagination caps again and think of
that in poison oak terms. Imagine that someone nonchalantly rubs poison
oak on the toilet seats or countertops of your favorite local shopping
venue. That's the equivalent for pet-allergic people. (Well, replace
itchy eyes for assholes, and replace watery eyes and sneezing for
burning crotch.)
The end.
(For comedians, you always leave them wanting more; for periodic local color and complaints, you go out on a reference to festering groin itch.)