When you're looking at the map, selecting clients, if you see a picture of a cat then choose it.
At first it seems like a prank call but when you get back in the cab a cat is there. Treat it like a regular passenger and don't kick it out of your cab. Eventually you'll drive it to the train station (its destination) and the achievement will pop.
I doubt it makes a difference but during the chat I asked if it was running away from someone and if she was being overly affectionate. I then tried to guess the location they were headed to and I guessed Normandy. I kept selecting the option "Go on" when it was available and chatted to the cat the whole way.