Not in the UK. The etymology for "dado" I've found is,
"1660s, 'part of a pedestal between the base and the cornice', from the Italian dado - 'die, cube' also from Latin datum (qv). Meaning 'wood paneling on the lower part of a wall in a room' is by 1787." The upper part of the dado panel normally terminates with a dado rail or chair rail which is sometimes held in a groove.
Whilst there certainly were dado planes, these fell out of use in most workshops before WWII in the main (seee below for an example). In a few of the older (Victorian and Edwardian) joinery and carpentry books I have, including Tredgold's (1871), there is reference made to dado planes being used to cut trenches or housings and (cross grain) rebates, NOT dados, and even the two copies of Audel's I have (an American trade series - mine are 1923 and 1945 editions respectively) refer to the joints as housings, grooves, trenches and rabbets - but not dados - whilst White's of Paisley used to manufacture a specialist type of cross cutting saw specifically designed to cut housings across planks (to manufacture low cost stock shelving) as a "dado or variety saw" specifically because it was designed to utilise "trenching or dado heads". Most of the Wadkin literature I have been able to find on the subject, including a late 1920s tooling catalogue, refers to the tooling as "trenching heads". They only started to use the term "stacked dado head" to refer to the stacked dado saw set when they started to make the sets in the 1950s, possibly as a way bto differentiate them from the heavy cast bronze split (2-part) trenching heads they had made up to that time and later:
(OK, this is a Robinson head, I believe in cast aluminium, but they were very similar to the Wadkin heads which I used on a crosscut saw "back in the day" - note the non-limiter design of the head which will drag you in if it catches you )
This leads me to suspect that the modern American use of "dado" is derived from a combination of the name for the cutter and a complete ignorance of the terminology traditionally used in the trade. I'd say that referring to these different cuts as housings (or trenches), grooves or rebates actually defines them better when talking to someone else than calling everything "a dado" which would leave the listener wondering if the speaker means a groove/housing/trench or a rebate...
A note about traditional dado planes: These were basically a fixed width rebate plane, often with a skewed cutter, and fitted with a depth stop and two nickers to score or cut the cross grain on either side. In use you fix a temporary fence to the workpiece and start by drawing the tool backwards to score (knife) the timber fibres with the nickers, then you proceed to plane the housing as normal, starting with short strokes at the far end and gradually increasing the length of the stroke until the full length of the housing is being worked. The plane is against the fence until you have a defined "track" which the plane will follow on its' own. There is no point in using a plane with nickers making cuts with the grain as there is no need to sever the fibres to make a clean cut - in fact doing so would probably result in any wild grain pulling the plane off the intended line.
Here is one (a traditional dado plane) from my collection:
Made by T. Clough of Bury near Manchester about 1828 (the only date known for him in business, so pushing 200 years old) and probably faster to use that a dado head - so long as you know how to tune the plane up (which is actually the hard bit)
