Full disclosure: I do have perfect relative pitch, and a kind of perfect pitch for a few notes. So your experience may differ from mine.
My recommendation would be to play with how chords are arranged. If you break apart, for instance, a C major with a minor 7th, you have 4 notes (generally speaking... piano experience here): C, E, G, and Bb. When played one at a time with each sustaining through to the end, you can clearly hear what notes are being added and then afterward hear what the full chord sounds like. So how I learned to recognize chords was to play them broken apart like this and recognize initially the increment from the 1st to the nth, and then just check pitch to tell whether the nth was higher or lower. With practice and repetition, this gets pretty fast, so that now I can hear a chord and am generally spot-on with what it is, and most of the time what key it's in.
As a side note, this makes it easier to tell instantly whether an instrument is out of tune, before any playing actually happens. Once you've trained your brain to listen for these things, just two notes played can give you a good idea of how well-tuned the strings may be.