That's the origin, ultimately. Not in Old Norse and its descendants, which borrowed it from Low German which had "Hantscho", meaning what you said. This survives more clearly in other languages, including German and Dutch.
Does Norwegian differentiate between gloves (with fingers) and mittens (without fingers)? Which one is hanske?
With Norway being cold and having a strong knitting culture, I assume they do have different words for them.
I wanted to know the same thing and typed "votter" and "handsker" into Google's picture search. It turned out that "votter" are a kind of mittens, obviously often knitted, and "handsker" have fingers and can also be made from leather.
"Jeg har hennes hansker" is accepted. The position of the possessive determines whether or not you should use the definite form of the noun. If it's before the noun it acts kinda like an article, so you won't need to use the definite form.