4K resolution
So while 4K UHD sounds like it has a pixel structures with 1/4 the area of 1080p, that does not happen with pixel shifting. Only a true 4K projector offers that level of resolution. That's why "true" 4K projectors cost so much more than 4K UHD projectors with otherwise similar feature sets. They produce smaller pixels, finer resolution, no compromising of detail or color from overlapping pixels. By comparison, the slight difference in aspect ratio between DCI and 3840×2160 pixel displays without overlap is insignificant relative to the amount of detail.
4K resolution
Pixel shifting, as described here, was pioneered in the consumer space by JVC, and later in the commercial space by Epson. That said, it isn't the same thing as "true" 4K. More recently, some DLP projectors claim 4K UHD (which the JVCs and Epsons do not claim).