I sing as an untrained amateur, and probably not too badly. The only "teaching" I ever got for many years was when a gr. 2 or 3 school teacher decided to give us solfege for that year, so I had a strong sense of diatonic patterns. Yes, it did help for many things. But it did not help for others, and the diatonic element even proved to be a handicap eventually.
My strength in singing helps me visualize a melodic line, and I often get comments that these come through strongly in my playing. That said, I'm overcoming a blurring of registers: I can easily play in the wrong octave and not notice. Even though I have (had?) 3 octaves, that still doesn't cover the whole span of the keyboard, and what we tend to do is to simply continue an octave higher or lower when music goes out of our range. That creates a kind of register-blindness.
I can sing chord types melodically; major, minor, diminished, augmented. But I am weak in recognizing the colour of certain kinds of chords - I might think a diminished chord is a minor chord, because I am hearing the minor third of two of the notes.
Another thing is that you should not "play like a singer". When I sing, I have to keep singing the note until it stops sounding. On piano, you may be holding that note with the sustain pedal, and move over early. Even if you "hold down" the piano key, you must release the pressure in your hand in order to gain relaxation - while as a singer your effort must continue - while on piano the strings resonate once struck without your effort.
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