All the important women in this film are rich, so we always see them in luxurious furs. Don't tell PETA. They live in New York City. It gets cold there. More than just looking pretty, the furs begin to represent a difference between New York theater actors (i.e. real actors) and Hollywood actors (i.e. terrible horrible no-good celebrities).
A Hollywood actress arrives at Margo's party, and we never even see her, but we do get to see her sable coat. Karen makes a smart observation, saying, "Women with furs like that where it never gets cold…" When Birdie comes to retrieve the coat, she has to ask which one is sable. To Birdie, Margo's assistant who doesn't have a fur coat, it's nothing more than just another fur coat. When Karen asks what she expects, Birdie says, "A diamond collar, gold sleeves —you know, picture people..." She doesn't realize she's not far off, considering all the outlandish outfits that show up on the red carpet...
When Margo announces that she and Bill are getting married, someone asks her what she'll wear. She says, "Oh, something simple. A fur coat over a nightgown." That's a loaded image. It contains glamour and sex, but also suggests that those fancy furs are covering up something much more ordinary. They're like that glittery trophy—they look good but don't necessarily mean anything. Underneath, even the stars are just real people.