This is certainly a good idea for a ground plane (also called a power plane). With sufficient stitching vias, two parallel ground planes results in lower volt drops at high frequencies and imporved performance especially with sensitive analogue circuits in the presence of digital circuits. Having said that, when you have mixed signals it will probably be better to keep a digital ground plane separated from the analogue ground plane to avoid circulating digital currents in the analogue ground plane. Stitching the digital and analogue planes at just one main area is also something to be recommended.
For non-ground power planes the benefits are less obvious. My general philosophy is that I make non-ground power planes sufficient for the current that are passing and use the "extra space" for more true ground plane. Using tracks (rather than planes) to feed individual chips can be regarded as a good idea because the track's self inductance (along with the chip's decoupler) make a nice power supply filter and you end up with a fair amount of filtering when this is extended to a multi-chip PCB design.
On the other hand, for low voltage MCU's (circa 1 V logic levels), it probably makes sense to keep a true Vcc power plane and having two (with stitching vias) probably is a good idea.
So many of these decisions are based on the target design and therefore generalizations are fraught with subtle dangers.
Doing so is good or bad for EMI/EMC?
Well, before you consider EMI or EMC, the PCB has to be fit for purpose and perform to the required expectations and that, is the motive to my previous paragraphs.When it comes to EMI or EMC it's ground all the way rather than a non-ground power plane.