A 'shaker kitchen' is the style of the door. It is, or should be wood where you have 2 vertical Stiles and to horizontal rails with an inset panel (98% of these are veneer) A shaker 'style' kitchen could be the real wood depending how the agents have marketed/worded the house or could mean it is a copy lookalike being a foil or vinyl wood with the same shape
It just normally means pale "painted" (not wooden) cupboard doors with an oblong frame and flat centre panel. About 15 years ago it was the bees knees to have a Shaker style kitchen and estate agents still mention it. But TBH it is a fairly classic style that hasn't dated.
when you search for "shaker style kitchen" click on "Images" and you will soon get the idea.
Shaker style in terms of kitchens just means a certain design of unit to me. Our kitchen has what are described as shaker style units (see profile). So door frame made of planks (or made to look that way), with recessed bit filling in the middle
It is a marketing gimmick! They should be very plain, very un-ornamental, but most things described as that are not anything a shaker would allow in their homes. They were very fashionable at the mid to end of the 1990s. Ignore the hype and see if it is well made and something you like, and make sure it has been fitted correctly.
And how they differ from other kitchen styles? The house we are buying apparently has a Shaker style kitchen, and I was trying to figure out what exactly a Shaker style kitchen is. Did a online search but couldn't find anything helpful.