It is hard to tell what animals 'see' when they perceive absolute pitch. At least in the case of bats it appears to be clear that absolute pitch cues are used for object texture discrimination. But this is a special case.
I think that for most animal species the memorization of a certain number of concrete sounds and their absolute pitches are sufficient for survival. Human language has provided a new situation. The variety of possible communication signals has made it necessary to extract more redundant, abstract features from the signals. Relative pitch is more robust with regard to the concrete sound source (e.g. different speakers) which makes it prone to the processing of more global stimulus features (the melodic contours in speech and – last not least – in music).
I completely agree with you that verbal labeling of the aboslute pitch categories is only one stage in the perception process. These labels depend on what you have learned when you were young. I see it more as a way to access the outcome of the absolute pitch processor. It would be interesting to know what are the labels the animals attach here. What do they imagine when they hear a certain absolut pitch object?
Do they "see" a big or small ape? or a "red" or "green" goldfinch?