Recently, Air India allowed the pet dog of a passenger who was flying in business class from Bengaluru To Delhi. The airline did not comment on it, however, an airline official said, "we allow pets on board (in passenger cabin). If the owner is in business then where will the pet travel? The final authority is of the captain (on allowing pet in cabin) and the rule is that the presence of the pet should not trouble other guests."
Though there is no rule about allowing pets in airplane, however, airlines generally don't allow them in passenger cabin. Indian airline Vistara does not allow pets at all. Initially Jet Airways used to allow pets in aircraft belly instead of passenger cabin. However, they stopped it after two puppies lost their lives.
Talking about International airlines, Emirates and Singapore airlines have strict policies against allowing pets in passenger cabins. According to the reports, allowing a pet in passenger cabin led to strange situations on an AI plane a year back. A source said, "The pet of a senior Indian official accompanied its owner to Europe about a year back. After a couple of drinks, the owner started walking the dog in the passenger cabin and took him to the toilet. He even asked the cabin crew to feed the dog and give him milk."
Under such scenarios, the airlines getting strict is seems valid.