Unfortunately, I have a great deal of experience with this bug. It appears to be connected to storing pets in a guild bank. After removing the pets from the guild bank there is a chance the caged pets will be converted to the bugged generic cages. It seems to be more likely the bug will occur if the pets are sent through the mail- I particularly encountered this after expired auctions were returned to me through the auction house.
I was able to have many pets restored to me by not clicking on any pet attachments in the mail that appeared to be affected by the bug (the icon will show a generic pet cage instead of the normal pet icon). The mail in this case should still include the name of the bugged pet in the title so a GM should be able to easily confirm what pet is in the attached cage with no danger that you could be lying about what pets were affected. It is a very time consuming process, however, for both you and the GM- I very clearly included in my tickets the nature of the bug and a list of every affected pet, which they could check themselves against my mail.
Since I figured out the nature of the bug I have stopped storing any pets in my guild bank and not had any further issues- pets stored in the individual character bank do not appear to be affected, so I would suggest storing your pets there instead.