Although there are a few diseases (such as rabies, ringworm and intestinal worms) that can spread between pets and people, most of them don’t cause respiratory symptoms such as coughing or sneezing.
And, in particular, pets do not appear to be susceptible to human rhinoviruses (the germs that cause colds).
Most contagious diseases are species-specific. This means that, for the most part, dog germs spread only among dogs, cat germs spread only among cats, and human germs spread only among humans. It is very unlikely that the dog you’re looking after caught a cold from you.
However, a nine-month-old dog who recently got lost and spent a day in the pound is in the prime risk demographic for a number of canine respiratory diseases including kennel cough. The stress of being lost and ending up in the shelter no doubt weakened her immune system. Close confinement with large numbers of other dogs in the pound likely exposed her to a variety of canine pathogens. Dogs less than two years of age are especially susceptible to the sorts of germs that are common in shelters. And the symptoms you describe are very consistent with kennel cough.
In my opinion, the Chihuahua you’re looking after picked up her cough in the shelter. You are not to blame.