Being a ward definitely doesn't imply a name change. You're not related at all. But Tim wasn't actually that much younger than Dick. He was still a minor where Dick was an adult, but we're talking late-teens. He was pretty independent by the time he was adopted. He was older than Jason, who was Jason Todd until (and, uh, after) he died despite being adopted by Bruce. Particularly after having Dick and Jason as examples I would probably have assumed Tim was still Tim Drake unless told otherwise. (I remember when Bruce gave Dick the papers as an adult he said made a point of saying he didn't have to change his name--it was pretty adorable.)
Not that Tim can't do that if he has his reasons. He has every right to the name just as Dick and Jason do and particularly in this story I think Yost has explained why it's important for him to use it. (I wouldn't be surprised if the arc ended with him going back to Drake because he doesn't need the name to tell him he's Bruce's son.) It's just that when Tim was adopted in canon it seemed like there were all these characters who suddenly were all "Tim Drake...or is it Wayne now?" as if it was a serious question both inside and outside of canon and that always confused me because I didn't get why I would assume the name was changed when Dick's and Jason's weren't and nobody ever asked why afaik.
Lucius here, of course, might have just discovered some shenanigans where money went into the account of a "Tim Wayne" and that's why he's storming around using the name.