It's complicated. DC's Captain Marvel started at another comics publisher, Fawcett, but for various reasons his trademark had expired by the 1960s, at which point Marvel made their own superhero named Captain Marvel (a name which was subsequently passed down to several other heroes, most recently and successfully to the former Ms. Marvel, Carol Danvers.)
DC then got the rights to the original in the 1970s, but since Marvel now had the trademark they couldn't name the comic starring him Captain Marvel, so they called it Shazam! after his catchphrase. This caused a lot of people to think the character was named "Shazam," so when DC reset all their books with the New 52 in 2011 they just officially renamed the character "Shazam." They seem to have backed off on that recently, and it seems the character is now going by "The Captain."
Anyway, the upshot of all this is that at the moment Marvel has a character named "Captain Marvel" and DC does not.
DC then got the rights to the original in the 1970s, but since Marvel now had the trademark they couldn't name the comic starring him Captain Marvel, so they called it Shazam! after his catchphrase. This caused a lot of people to think the character was named "Shazam," so when DC reset all their books with the New 52 in 2011 they just officially renamed the character "Shazam." They seem to have backed off on that recently, and it seems the character is now going by "The Captain."
Anyway, the upshot of all this is that at the moment Marvel has a character named "Captain Marvel" and DC does not.