Contrary to what some of the other answers have implied, there are actually a whole bunch of keyboard instruments with magnetic pickups, essentially analogous to electric guitar PUs. They all sound significantly different from their acoustic pendants, but that's in a sense a feature – some of these sounds have become legendary and now feature in almost any digital keyboard!
- The Clavinet is almost the same as the baroque acoustic clavichord (a string instrument with a different hammer mechanism from piano), and indeed in single notes you can hear the semblance – but while the clavichord is infamously quiet and only useful for very delicate chamber music, the Clavinet benefits from the amplification option and has become famous for a relentlessly bright, highly rhythmic sound, best associated with funk songs like Stevie Wonder's Superstition.
- The two most famous models of electric piano, the Rhodes and the Wurlitzer, both use magnetic pickup. However they do not have strings – the vibrating elements are solid metal plates. A Rhodes is actually rather an electric Celesta, rather than -piano, and that's the main reason these instruments sound very different from piano.
- The Yamaha CP-80 (and its siblings) is an almost fully-featured grand piano, but built with the intention to be played amplified with its magnetic PUs, not acoustically.
These sound much closer to an acoustic piano than Rhodes or Wurlitzer do, but still strangely sterile in the dry signal. Not too surprising – a completely clean electric guitar also doesn't have the depth / resonance of a mic-recorded acoustic guitar.
- The Hammond organ uses magnetic PUs on tone wheels. This does not have an acoustic pendant.
In summary, it is absolutely possible to use electric-guitar-style pickups on non-guitar string instruments, but it doesn't make a lot of sense to retrofit an acoustic instrument with such PUs. They are mostly useful for processed, loud, possibly distorted sounds, in which case it anyway sounds basically like a different instrument. Indeed, even the “standard clean” sound people associate with the above instrument is actually result of amplifying them with a nonlinear, possibly tube amp without HF tweeter – again, like you do with guitar.
The only requirement for magnetic PUs is that the strings (or otherwise vibrating elements) are magnetic – which of course isn't the case for example for harps! However, instruments with non-magnetic strings could generally be redesigned to use steel strings, though that might again make a lot of difference to playing feel and certainly to the sound.
Piëzo pickups tend to reproduce the acoustic sound more closely than magnetic ones do, as well as not making any demands on the string material. Contrary to popular opinion the characteristic “magnetic sound” actually has less to do with the magnetic design per se, but rather with the fact that such PUs typically have very many windings, causing a high inductance which together with a passively-driven cable to the amp makes for a resonant low-pass filter. An active magnetic PU can be designed to reproduce the original spectrum quite well, though it will not pick up much of the body resonances. Therefore, to get an acoustic-like tone, you'll always need to combine it with reverb and/or additional microphones.