You should get back in touch with KLM - the person you were talking to may have confused the animal travelling as excess baggage as opposed to cargo (essentially the same thing - the animal travels in the hold, but is in a specific category = cargo).

I have travelled 4x with my pets with KLM and cannot recommend them highly enough. The animals are held at Schipol for the stopover, seen by a vet, watered and walked (or litter boxed) and I'm sure petted over by the staff at the Pet Hotel there. The flight crew has always been very helpful in confirming and reassuring me that they were loaded on board and were doing well - even the ladies at the information booth in Schipol allowed me to speak with staff at the Pet Hotel to reassure me my babies were doing OK.

The last time I did this was several years ago and things may have changed - my pets flew on my flight but were classified as cargo - so hopefully it's just been a misunderstanding.