It's the same here up north of the border, and sometimes we hear tote or satchel once in awhile, or just bag. Handbag is pretty common, especially for Commonwealth countries.
My boyfriend is originally from Maryland and when he called my purse, "pocketbook" I was and it made me chuckle. I'm originally from Texas and use the term 'purse'. I
I've always called it a purse and am originally from the midwest. Some of the girls that I teach with from NY call them pocketbooks. Crazy! I rarely hear anyone call them handbags here in Atlanta.