One experimental method which does not involve naming pitches is to ask people to sing well-known songs which they most likely "learned" by listening to the original recording (e.g. "Waterloo" by Abba, "Imagine" by John Lennon, etc.) and compare their pitch with the original.
See https://pdfs.semanticscholar.org/f977/811c9023707817eb026a9710b047a115a3a5.pdf for such an experiment, and http://www.doc.gold.ac.uk/~mas03dm/papers/Frieleretal_LevitinReplication_2013.pdf for a replication of it.
One conclusion from the original study was that accurate pitch memory is fairly common (40% of the random experimental subjects demonstrated it to some extent) but pitch naming is much rarer - and as the OP says, the "naming" aspect must be learned behaviour, not innate.