I'm not familiar with the minutiae of of piano tuning, but nothing prevents you from tuning your piano keys so that F# and Gb are two different keys. The only reason why they're the same key is because we use an equal temperament tuning. Of course, if you tune the flats and sharps separately, then the range of your piano would be considerably reduced.

And yes, fretless instruments such as violins can be precise enough to differentiate between the two.