Which sentences sound idiomatic to the Americans:
I am a big Elvis fan.
I am a big Elvis admirer.
I am a big Obama fan.
I am a big Obama admirer.
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American here. "Fan" sounds more idiomatic to me, but both are common.
In informal situations, I think "fan" is almost always used. "Admirer" is a bit more formal—I would be more inclined to write rather than say it.
There's also the idiomatic phrase "secret admirer" which is commonly used among children to refer to a person with a secret attraction to someone.
It might be helpful to know that "fan" is derived from "fanatic", meaning someone who is interested to the point of obsession. Sports fans are a great example of this kind of enthusiasm, willing to make great personal sacrifices to support their team. Popular works of fiction like Harry Potter or Star Wars also have very devoted fans.
There is an implication of (at least) mild irrationality when describing someone as a fan. Performers and public figures will often express gratitude to their fans, saying "I'll do anything for a fan!" or "This is for the fans, you make all of this possible!" Some fans are more casual, but all consider themselves more invested than a typical person should be.
Conversely, an "Admirer" is generally very reserved in their admiration. An admirer is not going to dress up in a costume and stand in freezing weather to support their team - a fan might.