Cats do not become ill from to the human Influenza Virus (flu), but can become infected with other feline viruses (Calicivirus, Feline Herpesvirus, Rhinotracheitis), causing illnesses which have been referred to as "cat colds" in the past. These cat viruses are distinct from human Rhinovirus, or cold virus to which cats are also not susceptible. A so-called cat cold is not the same illness as a human cold and for this reason the use of the term is now discouraged. Like cats, dogs are not affected by human flu and cold viruses, but have their own set of viruses (Parainfluenza and Adenovirus) which exhibit similar symptoms.
Sneezing, sore throat, and nose and eye discharge may be related to infections of the nasal cavity, sinuses, and eyes caused by all types of infectious agents (bacteria, virus, or fungal), so laboratory tests may be required to identify the culprit.
Note that the streptococcus bacteria group is found in humans and in the canine and feline species. In pets, this germ sometimes causes sore throat and other symptoms including weakness, fading puppies and kittens, tonsillitis and shock. Note that dogs and cats can carry the strep type A bacteria without showing signs, and are thought to be a source of human infections. Humans, however, are not thought to be a source of infection for pets. When conducting a diagnostic evaluation of a sick dog or cat, a veterinarian will consider bacterial infections as another possible source of illness.
Dogs and cats that are "under the weather" should be seen by a veterinarian for a professional assessment. The veterinarian may recommend laboratory tests, and will prescribe supportive therapy for a patient with viral infections causing flu-like or cold-like symptoms.