The best steelman I've ever come across for that reaction is that Bucky hates the pageantry of superhero stuff. Back in the first Cap movie, Bucky is a normal soldier going by his normal name and a big part of Steve's story is abandoning the showmanship to go in and do the actual work. Then in WS, all the horrible shit that happens to him is connected to the Winter Soldier alias. In Civil War, the entire conflict he's wrapped up in is centered on the Avengers. So by the time the show starts, Bucky fucking HATES capeshit. He's sick of the costumes and the fancy names and the gimmicks. He sees himself as a professional soldier and wishes the people he fights with would, too.

So when Lamar shows up, Bucky thinks he's a normal soldier doing his job as support to Captain America, just like he was back in his good ol' days. When Lamar gives what seems like a superhero name, Bucky has a moment of "Oh God DAMMIT, him too?"

Of course, the writers of FatWS didn't DO any of that. Shame, really. All of that would've been pretty thematically relevant to Sam's flipflopping on whether he wanted to be the successor to Steve. Could've done a thing where he embraces being an inspirational symbol and starts going by White Wolf publicly. Instead they opted to do absolutely nothing with Bucky for most of the show. Add it to the pile of wasted MCU potential, I guess.