Ok, so Duden threw me off:
That's always the danger when you try to deduce the meaning of words solely by using a dictionary and not from experience.
That being said, Duden is not always the best source for colloquial or slang. Try using Redensarten or Mundmische. Da wird 'abschleppen' ziemlich eindeutig beschrieben.
In my own experience, abschleppen for people (Mädchen abschleppen) is "to pick up (women/girls)"; abschleppen with things is usually used with cars - to tow (away) a car.
The AE "to pick someone up" can mean both abholen and abschleppen, depending on context. I think that's what caused the initial skepticism with some native German speakers in this thread, albeit 15 years ago!
4. To take on (passengers or freight, for example): The bus picks up commuters at five stops.
7. Slang To make casual acquaintance with, usually in anticipation of sexual relations.
The American Heritage Dictionary of the English Language, Fifth Edition