Here is one of country's finest violinists Joshua Bell playing for free at Washington DC two days after his show sold out at $100/seat in the same city. See what happens:
Over 1000 people passed by, but just 7 stopped by to appreciate the music, and just one recognized him! Joshua Bell was doing exactly what makes him "Joshua Bell". But without the name "Joshua Bell", he was just an ordinary street performer. He didn't get any real world "followers"!! Thinking of some celebrities, Lady Gaga doesn't sing on Twitter, Shaquille O'Neal certainly does not play basketball on Twitter, and Julia Roberts doesn't act on Twitter.But still, each of them is a Twitter-celebrity too! In fact, Julia Roberts doesn't even tweet on Twitter!! With 0 tweets, she has 300,000 + fans on Twitter!
My point here? Celebrities reach their celebrity status with a lot of
hard work, talent, and some luck, but once they attain that status,
people follow them mostly for their name, not necessarily for what they are
known for. Sometimes, it's the desire to peek into celebrities' life; sometimes, it's the illusion of being connected to a superstar. But, in the end, it's closely tied to their identity as a celebrity. Thus, to answer your question, most celebrities would drastically lose followers if their identities were hidden.
1) Please note I said "most celebrities". There are definitely some celebrities and semi-celebrities, especially from literature and journalism fields, who do tweet good content, and they may continue to have decent following even if they consider writing under a pseudonym.
2) If you are interested in the Joshua Bell experiment, you can read full story here:
See question on Quora