The OP has a very strong point in criticizing singers singing out of style. Record companies sometimes indulge in crossover experiments with famous performers, aimed at their admirers. Barbra Streisand singing Brahms, for example, was a universal ridicule. But even within each of the major categories such as classic, pop, jazz, and folk, there are hosts of stylistic approaches, each requiring good acquaintance and special training, as Will exemplified in the classical case. Sadly, many attempts fail, and more sadly still, this often goes unnoticed by the public, so that whole genres are misrepresented and brought into disrepute.

I'm with Gerry et al. that a good general training will mostly be found useful, if it is complemented with the requirements of the specific style of the pieces to be performed.

Shakespeare, on the other hand, was and is Shakespeare (whatever his or their real name/s may have been).