Is there an in-world reason all of Dick Grayson’s successors accept being called Robin? Why didn’t Jason, Tim, or Damien say “Sure Bruce, I’ll fight crime with you, but if I’m going to be named after a bird it’s going to be something cool like Falcon, not some dumb songbird that goes tweet tweet.”
They considered a badge of honor? And more of a transferable title than a name?
Batman is NOT going to be overshadowed by an assistant with a cooler name.
Robin did change his nom de héro to Nightwing.
Well, Dick Grayson did, when he ceased being Robin. The OP’s asking about Grayson’s successors to the role as Batman’s sidekick.
Because the brand is “Batman and Robin”. I mean, who the fuck are “Batman and Falcon”?
Also, “Falcon” is easily turned into something like “you Falcon idiot!”
Not to mention being the pre-existing name for a superhero for the other major comic company. 
Sure, branding is important, but sometimes you’ve got to update your brand name and logo to stay current.
Batman and Barn Owl would make a bad-ass crime-fighting duo.
Batman and Sea Cucumber…not so much.
But Aquaman and Sea Cucumber…watch out!
Tim did go by Red Robin for awhile. I’ve always kind of thought of the time working with Batman as the Robin years, and they only “earn” their own hero name after they’ve spent enough time being tortured by Bruce.
True, Sea Cucumber could help Aquaman escape precarious situations by eviscerating his intestines…but, I’m not so sure that would make good cinema.
I’d just suspect they aren’t in the position to make that demand. Dick was an established fighter who goes off to work for himself, so he can change his name. But, if you want to work with Batman, it seems you have to be Robin.
I could see Damian complaining about the name, though. At least, based on the Battle of the Super Sons movie.
Who?
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I’d just like to note for the record that Robin was not originally named after the bird, but rather in honor of one Robin of Locksley, an early proponent of the redistribution of capital - which in retrospect, makes his association with a certain billionaire playboy somewhat ironic.
Batman and Blue-Footed Booby has a nice ring to it, while sticking to the avian theme. And, for reasons I can’t quite comprehend, it has a somewhat titillating vibe to it.
The role each of them signed up to fill is Batman’s sidekick “Robin.” If they don’t want the role, they can go do their own thing in their own costume - it’s what Batgirl did. Jason and Tim (and later Carrie Kelly) were looking for that specific job.
The only person I can see considering a new name would be Damien, and he’d have gone an entirely different route if he had. As he grew to admire Nightwing/Dick Grayson, he became more disposed to being Robin.
Hence the Olde English way his name was spelled out in the titles.
In grittier, more cynical versions of the property, the answer is something like “because they’re nameless and faceless, existing only to sacrifice and eventually be killed or crippled in service to a mad vigilante.”
It’s marginally less obvious when Bruce Wayne gets a new “ward” but Batman still has a masked kid named Robin. If new kid turned up the same time as new sidekick the cover story gets really weak.