Musk later emailed the BBC, saying: "We are aiming for maximum transparency and accuracy. Linking to ownership and source of funds probably makes sense.
"I do think media organisations should be self-aware and not falsely claim the complete absence of bias.
"All organisations have bias, some obviously much more than others."
"State-affiliated media" refers to publications where the government "exercises control over editorial content through financial resources, direct or indirect political pressures, and/or control over production and distribution," according to a page in Twitter's help centre that is linked from the label on the BBC account.
The phrase was also used to describe the Twitter account of US broadcaster NPR, although it has now been modified to read government-funded media, the same as the BBC account.
The BBC has consistently upheld its objectivity, and it functions under a Royal Charter that was negotiated with the government and states that it "must be independent."
The corporation's production is paid for by annual licence fees from Britons of £159.
The government decides the amount, but private households pay it.
While the BBC's main account, which has 2.2 million followers, was labeled, far larger accounts linked with the corporation's breaking news and sports programming were not.
The account primarily tweets about BBC-produced television series, radio shows, podcasts, and other non-news content.
It comes after Twitter removed the blue verification mark from The New York Times' account.
