These types of phrases and slang can vary from country to country, city to city and from one generation to another. In my circle of influence, people would tend to say, "...strange looks FROM other women shoppers". But this does not mean the other way is incorrect. It's probably less common, but not necessarily incorrect. Also, this is a script from a book or TV show, then it is a quote from one of the characters, and people do not normally speak using proper grammar.

The words are not really interchangeable. The word "from" is much better.

Gee...I can't decide which one is best answer. Each of you has a perfect, specific, clear explanations! Thank you all. Reallllly apprecaite your helps.

I agree, "strange looks from other women shoppers" sounds better, but "off" can be acceptable in some situations, for example :- "I got 50 dollars off him." "I managed to get a jumper (sweater) off him." I guess "off" is used when a specific effort is required to remove a physical object from another person. The effort which got "the strange looks" was not specifically made to obtain these "strange looks"

From is much more natural English. However, when I read it, I understood the text saying "off." The terms "bouncing off" and "reflecting off" are used in certain situations where "bouncing from" and "refecting from" would not be considered natural. In basketball, you bank a shot off the backboard (NOT "bank from the backboard.") In this case, saying "strange looks from other women shoppers," works the best.