Not a complete answer to your question for reasons mentioned below, but you could start by doing a few experiments to find out why it is in your case that sung notes don't trigger the automatic verbal associations (note names) which appear in your conscience like when you hear a piano note or other instrument.
You could, for example, take some recordings of sung notes, then, using Autotune or Celemony Melodyne or other such software:
- Get the notes tuned perfectly (so they are bang on the right frequency for whatever the note is).
- Remove pitch wobble in the singers voice.
- Remove natural vibrato (if it has any).
Once you've done that, see if you are able to recognize the note. If you can, you'll then know that one or more of those things has something to do with it. If it does, you could then try to train your existing AP to recognize notes with ever increasing amounts of wobble and vibrato by taking the idealized notes you created then re-introducing a tiny amount of vibrato/wobble/etc. back into the notes, testing yourself again, then gradually introducing more and more back in until they are back to natural sung notes again. Might work in your case. Would be interesting to hear if it does.
AP can be improved and developed to some degree using different methods (see article below), but nobody really knows at present what methods can create and maintain verbal/tonal associations of the strength and accuracy of those who have 'natural' AP which they developed in childhood during the hypothesized plasticity period. Presumably your instrument mediated AP is of that sort.
From 'ABSOLUTE PITCH LEARNING IN ADULTS. Is it impossible to acquire absolute pitch in adulthood?' by Yetta Kwailing Wong, Kelvin F. H. Lui, Ken H.M. Yip, and Alan C.N. Wong:
In contrast, while AP can improve to some extent in adulthood with
deliberate practice (Brady, 1970; Cuddy, 1968, 1970; Hartman, 1954;
Meyer, 1899; Mull, 1925; Russo, Windell, & Cuddy, 2003; Van Hedger,
Heald, Koch, & Nusbaum, 2015; Wedell, 1934), there is no convincing
evidence that adults can attain a performance level comparable to the
AP possessors through training (Bachem, 1940; Levitin & Rogers, 2005;
W. D. Ward, 1999).