“His father had always been an urban dweller(…) nipping through the streets by taxi(…)”
The Oxford Dictionary mentions “dash” as a possible synonym (among many different meanings).
Does “nipping” stands for “dashing”, in the event?
Yes. "Nipping" and "dashing" in this context are fairly interchangeable. I would hasten to say these are probably limited to British English, though I'm not qualified to say if they go further afield. They are on the informal side of formal (if you understand what I mean), whereas their cousins "tootle" and "pop" are certainly informal.
EDIT: For clarity, "nip" and "dash" most definitely exist in other variations of English, but not in this context perhaps. Used as in the example, they both express a swiftness or simplicity.
Where are you off to? Oh, I'm just nipping down the shops for a loaf.