Are you completely sure. I'm checking my students writing and I don't know myself what is right. I would have write the first option myself, but looking on the net I didn't find any sentence like the first one, he second one seems to be common, the thing is that I don't understand the grammatical reason for that.
This are exemples I found in the net:
"David Cassidy is a singer and actor, most notably appearing as one of the stars of the musical television series The Partridge Family in the 1970s."
"Cher is an American singer and actress who got her start as half of Sonny and Cher in the 1960s."
Hmmm, well, ok-- If I was writing the exact phrases you gave at the beginning, I would write the first option because it sounds more precise, but if I was checking students work, I would not mark the second option as wrong or grammatically incorrect. The examples you quoted are perfectly fine. I don't have a grammar book rule to quote to you, but in general my gut instinct is that as a standalone phrase, I would prefer "She is a singer and an actress," but if I continued the sentence with more information, in many cases I would probably write "...is a singer and actress..." depending on what followed.