Of course everything depends on where you live but - where I live (Chicago, USA) - “wardrobe” is not commonly said.
A CLOSET is a small space (it can have a door or just be open) where you put things like clothes, towels, coats, etc. Someone can have a closet in their bedroom where they hang all of their shirts or dresses. Someone else can have a closet in their home where they keep all of their winter coats. Someone else can have a closet outside of their bathroom where they keep all of their folded towels and other toiletries.
DRESSER is the word that we tend to use in place of wardrobe. A dresser is a piece of (usually wooden) furniture that has multiple drawers that you can open and close. You put your folded clothes in these dresser drawers.
A WARDROBE by definition is basically a “dresser” that also has a section to hang some of your clothes. Even though I know the word “wardrobe” and what it is, I’ve never heard anyone say it. We usually just say “dresser” 😆
🌈🫶🏼 Nicky