Oh, I have no sympathy for DC, I mentioned their shady as hell behavior, as a fan of comic history (in particular golden age stuff) there's just a certain sadness to the fact that a character that was once more popular than Superman can't be known by the name he was first published under because it got caught between the two rival giant companies.
Marvel jumped on the name when it became available in, what I think is fair to say, a move done almost entirely for legal rather than creative reasons. They didn't want anyone else to be able to slap the name Marvel on a cover (though they also did similar things with other once popular characters whose rights had lapsed, like Ghost Rider). Marvel was always, for whatever reasons, really resistant to the idea of 'promoting' Carol even though she'd been more popular than Mar-Vell and even had her powers stolen and wrote her off for a long period of time.
For the record, Billy was indeed called Captain Marvel within the TV show, but it had to be marketed as Shazam because Marvel Comics owned the name for promotional purposes. DC has also converted to calling Billy Shazam within the comics back in 2012, because of the confusion over the name issues.
No one should really be angry over it any of it in 2019, but there's a lot of interesting history and a lot of characters that have been known by the Captain Marvel name (and that's before you get into the Marvelman/Miracleman issues from the UK), so I don't think it's just a simple situation where Carol is the first name that pops into everyone's heads when they think of Captain Marvel, it might be Billy, Mar, Monica or Genis for a lot of people depending on when they first became interested in comics/super heroes.