(You could replace 'Fan' with words that are 'more german' like 'Anhänger', 'Bewunderer', ... but I think 'Fan' is the most widely used term for 'fan' also in german speaking countrys.)
Regarding persons you use normally "Fan" - except it is e.g. in relation to religious leaders where the meaning is a different one of Anhänger - e.g. Die Anhänger (meaning followers) Jesus.
Replacing Fan with Anhänger regarding actors etc. sounds weird in my opinion.
But Anhänger is as a perfect synonyme for fan of e.g. sport clubs.
