The tutor I had learnt briefly from a couple of years back used to occasionally ask me to sing (la-la, or aa-aa) along with him playing some scale like movements - probably major scales in chromatic progression. I used to find it interesting. Is this a useful and prevalent practice? If so, what are good vocal exercises for pianists?
My guess is that this was an ear training exercise. I use exercises similar to this with my students where I have them sing an interval before playing it.
This is to try to train them in the process of improvisation, where it is important to think the notes in your head before playing them. If they can sing this aloud before/while playing, then theoretically they can "sing" a musical thought in their head and then play it.
With that in mind, singing scales, arpeggios, and intervals as you play will help cement them into your brain. From that point, move to trying to sing them before you play them, and then singing a familiar tune as you play, and finally singing an original tune of your own as you play.
I might ask an instrumentalist to sing as a demonstration of how a phrase or rhythm OUGHT to go, as preparation for training the fingers to actually perform it! Or maybe as practice for the 'aural tests' component of a Grade exam. 'Singing back' a phrase played to the student can be a requirement. Some students can find this intensely embarassing.
Or maybe this teacher just thought singing was a basic musicianship skill that should be encouraged. Perhaps as well as teaching piano they ran a choir, and was always looking for recruits - singers who already read music are at a premium!
I wouldn't think particularly of 'vocal exercises for pianists', just vocal exercises in general. Yes, scales, arpeggios etc. And plenty of 'sing what you see' from notation.