I don't have perfect pitch, but I do sometimes experience something that seems to resemble it. For example, I often know if music is in D (or D minor). I associate something that resembles colour to C minor and E flat major (but different colours). If a pitch appears green, it's always E. It's a different type of perception for each key/pitch.
When I play music, I often think about whatever I was thinking about when I learnt it. It might be a discussion, for example.