Batwoman was already an established hero in Gotham from 1956 to 1964. And yes, for the last three of those years, she was joined on occasion by Betty Kane’s Bat-Girl, but the younger heroine was far less of a staple of Silver Age Bat-Books. And while Barbara Gordon was far closer in personality and appearance to Kathy Kane than she was to Betty, she was still not all that keen on claiming the Batwoman mantle from the more established superhero.
I would recommend reading Batman Family #10 for more information.
well some of it is sexisms’, Batgirl was also introduced in 1967 however Kathy Kane aka the original Batwoman was introduced in 1956 and was still somewhat active at the time and Babs was actually introduced in detective comics #359 a few months before her premier in the tv show. as there was already a batwoman in the comics, she became Batgirl. Current cannon has her as Batgirl starting at fifteen or sixteen then at 18 she gets shot by the joker in the killing joke and batgirl was gone from 1988-1999 outside the Batman TAS continuity until 1999 when Helena Bertinelli first usurped the title then Barbara Gordan passed it on to Cassandra Cain. since the new 52 she was seventeen when she got shot, and took up the mantle again about a year or two later when she finished her PT. and Batwoman/Kate Kane may have been working by then, and she was already used to calling herself Batgirl. I am not actually sure timeline wise she takes up the mantle. however now she goes by Oracle having passed the mantle of batgirl once again to Cassandra Cain.
However in The Batman Cartoon Batgirl does try to go by Batwoman, however she is both short, and of indetermanate age but younger then B:TAS and Batgirl sticks to her.
Usually because continuity keeps them young, but a lot of characters do “mature out” of their kid/boy/girl/lad/lass names. Sometimes taking over for their fallen/retired namesake, sometimes to overtly say “I am no longer a kid and my name should reflect that,” and sometimes just because continuity is weird and things get revamped by different writers. (e.g. modern Babs having started much younger than classic Babs.)
Kid Flash → Flash
Bat-Girl → Flamebird / Hawkfire
Batgirl → Oracle
Beast Boy → Changeling
Aqualad → Tempest → “just Garth”
Aquagirl → “just Tula”
Wonder Girl → Wonder Woman (x3)
Wonder Girl → Troia
Elasti-Girl → Elasti-Woman
Star-Spangled Kid → Skyman
Girl 13 → Traci Thirteen
Hawkgirl → Hawkwoman (pre-Crisis)
Sandy The Golden Boy → Sandman
Doll Girl → Doll Woman
In the Titans of Tomorrow pre-Flashpoint future:
Star-Spangled Kid → (Stargirl) → Starwoman
Aquagirl (Lorena) → Aquawoman
Most of the Legion I’m no LoSH expert, but know of at least a few…
Chameleon Boy → Chameleon Man → Chameleon
Color Kid → Color Queen
Cosmic Boy → Cosmic Man
Ferro Lad → Ferro
Lightning Lass → Pulse
Saturn Girl → Saturn Woman
Star Boy → Starman
Lightning Lad → Lightning Man
Phantom Girl → Phantom Woman
Spider-Girl → Wave
Shadow Lass → Shadow Woman
Bulletgirl → Bulleteer (different characters, but clear evolution)
I think this one needs an out-of-universe answer. Yes, it totally makes sense if adult woman Barbara were a real person to want a mature name, and since there’s already a Batwoman, she’d need to come up with something else, maybe not even bat-related. In-story, remember, the “Batgirl” is a playful costume she whipped up for a masquerade, and blurts it out in-the-moment to protect her identity - so she never really sat and pondered. But out-of-story, she was created to appeal to (younger) female viewers of a TV show, which was already very successful with “Bat-branding” everything, so it makes sense to stick with that. (Would “Batlady” or “Batqueen” or even “Batwing” (“The Batman Nobody Knows” wouldn’t come along until 1973) have worked just as well? Probably. But they went with Batgirl. It’s certainly catchier, and has precedence, if you leave out all the maybe-sexism-but-maybe-not logic…) Once she’s named, as @CaptainYesterday mentioned, the marketing money is pumped in to the name, and it’s very hard to change it. (Many characters have changed names, but once you’ve got public recognition, it’s tougher.) Reboots give writers a chance to retcon in a better explanation for why an adult woman is calling herself a girl.
This is true even without the TV link – new characters who are introduced under a protégé or in tribute to another character with name recognition have consistently sold better than those who are introduced fresh. Less people are going to pick up “The Librarian”, even if it’s exactly the same story and art that’s published under “Batgirl”, which is likely going to be sold on the shelf right next to the “Batman” title they came in for.
I will say that I wish that she had taken on the Batwoman name circa 1980. Kathy Kane died in 1979, and the start of a new decade would’ve been the perfect excuse to let Barbara “grow up.” Instead, Babs became a bit of a third wheel in the 1980s (when she was equally important as Robin throughout the 1970s), and the ultimate result of her sidelining was The Killing Joke, where she seemingly represents the death of a more innocent era in comics.
Well, I believe Babs was semi-retired by the end of the silver age and working as a congresswoman. If the DC database is right she only put the costume back on during Infinite crisis. however, the death of Jason Todd (same year) also helped put that nail in the coffin of the innocent era…and the end of batman getting to be somewhat happy in the mainstream.
She was leading the series Batman Family during her time in Congress. Batwoman died in Detective Comics #485, and Barbara lost her reelection campaign shortly afterward in Detective Comics #488. The latter comic came out in February of 1980. She remained a fairly regular backup character in 'Tec until the end of 1982, and then she was used only infrequently after that.