Can people with absolute pitch identify the exact frequency, or simply the pitch class? Absolute pitch can be defined as the ability to identify or re-create a given musical note without the benefit of a reference tone.

When people claim to have absolute pitch, does this mean they have the capacity to detect the exact frequency of a note, or merely the pitch class (i.e. C,C#,D, etc)?
Short answer
In practice, absolute pitch is generally tested for by using musical pitch classes.

Background
Absolute pitch (AP) is the ability to identify the pitch of a musical tone, or to produce a musical tone at a given pitch without the use of an external reference pitch. Most humans process musical pitch relatively rather than absolutely, and in fact most sensory systems operate relatively and not absolutely. AP is indeed a rare ability, occurring in less than 0.01% of the general population (Tekuechi &amp; Hulse, 1993). 

The way in which AP is tested is mostly using pitch classes. Pitch classes are likely preferably used experimentally, because people with AP are often musicians. Musicians are already familiar with pitch classes because of their training. In turn, AP typically manifests itself in trained musicians (Zatorre, 2003). Therefore, testing AP using frequencies expressed in Herz may simply be too abstract for most people with AP, since they are more accustomed to the use of pitch classes.

Note that, because the pitch A4 used to tune orchestras, corresponds to 440.0 Hz by modern convention, AP as tested with pitch classes can be, indirectly, considered to represent frequencies. 

Some subjects with AP may consistently identify pitches a semitone low or a semitone high. This can occur because a subject may be accustomed to a different tuning standard for A4 than 440.0 Hz (Tekuechi &amp; Hulse, 1993). Hence, AP has a strong correlate to training.   

References
- Tekuechi &amp; Hulse, Psychol Bull (1993); 113(2): 345-61
- Zatorre, Nature Neurosci (2003); 6(7): 692-5