The broadcaster says it is speaking to the social media company about the designation on the BBC account.
It said in a statement: “The BBC is, and always has been, independent.
“We are funded by the British public through the licence fee.”
In an email to the BBC, Elon Musk said: “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.”
He continued: “I should note that I follow BBC news on Twitter, because I think it is among the least biased.”
The label links through to a page on Twitter’s help centre which says “state-affiliated media” are outlets where the government “exercises control over editorial content through financial resources, direct or indirect political pressures, and/or control over production and distribution”.
That label was applied to US broadcaster NPR’s Twitter handle, but it has been changed to government-funded media – the same as the BBC account.
The BBC has always maintained its impartiality and operates through a Royal Charter agreed with government, which says it “must be independent”.
Britons pay a £159 licence fee each year to fund the corporation’s output, which is set by government but paid by individual households.
While the BBC account, which has 2.2 million followers, has been given the label, much larger accounts associated with the corporation’s breaking news and sport output are not currently being described in the same way.
The account mainly tweets about BBC-produced TV programmes, radio shows, podcasts and other non-news material.
Twitter has been contacted for comment.