Here's why Alfred always refers to Batman by the old-fashioned term "Master Bruce." While he may not don a superhero costume, Batman's butler Alfred Pennyworth is still a beloved part of the comics. Alfred made his debut in 1943 where he was treated as comic relief. Alfred was originally conceived of as a bumbling, wannabe detective, but his comic side would be toned down in later appearances.
Just like the Caped Crusader himself, Alfred has been played by many different actors across live-action movies, TV shows, and video games. Arguably the most famous portrayals have been Michael Gough, who first appeared in Tim Burton's Batman, Michael Caine in Christopher Nolan's Dark Knight trilogy and Efrem Zimbalist, Jr. in Batman: The Animated Series. Jeremy Irons would play the character the DCEU movies, while Sean Pertwee became a fan favorite for his take on Alfred Pennyworth in TV series Gotham.
Alfred Pennyworth's origins have been changed a few times throughout the Batman comics, though most depictions portray him as a surrogate father to Bruce Wayne after his parents were murdered. This led to a touching father/son bond between the pair, with Alfred typically referring to Batman as either "Master Bruce" or "Master Wayne." Since Alfred is a trained butler, etiquette dictates he would have referred to young Bruce Wayne as "Master Bruce," and his father Thomas as "Mister Wayne."
It stands to reason Alfred should have started calling Bruce "Mister Bruce" when he became an adult, but perhaps out of respect to his father he still uses "Master Bruce." This could also be a sign of affection for Bruce since Alfred has raised him from a young age. In formal settings, Alfred will refer to Bruce Wayne as "Master Wayne" but in private he will revert to "Master Bruce."
Batman never seems to challenge him on this point, so he clearly has no issue with being dubbed "Master Bruce." While Alfred isn't likely to wear a cape and swing across the rooftops of Gotham City, the comics have shown he's pretty capable of putting up a fight and has saved "Master Bruce's" life on more than one occasion. Other depictions have shown a more grizzled, combat-ready depiction of the butler, including Sean Pertwee's Alfred in Gotham or the version of Alfred seen in short-lived animated series Beware The Batman. The Pennyworth TV series also follows the gritty younger adventures of the title character in 1960s London.