| I am not sure what "logic" you are referring to.
It's simply the rules set by the HKAFCD.
You should look up the definition of an airway bill.
If you traveled on any of the major US airlines, you have NEVER received an airway bill for your pet. It is only produced when an item (in this case pet) is transported as manifest cargo.
Perhaps the rules change since you had traveled or the rules were not strictly enforced, but current office strictly declares this is a requirement. | 
