Earlier quoted context omitted.
kg/N varies, but by less than your normal margin of error. That's vastly different from deciding to use newtons to weigh "kilograms indicated" on your trip to mars when they give a vastly different number.
My point was that we don't weigh ourself in 'newtons' despite that being what we are actually measuring. We make some reference conversion to a much more intuitive unit, kilograms. We rely on the fact that it's a pretty constant conversion between the two. Is this really much different than that? I don't think so. The aerodynamics of the plane (stresses on the structure, performance of the aircraft w.r.t. stall, etc.…
Accurate weight-measuring scales rely on m = F_w/g, where m is mass, F_w is the normal force delivered by the scale's measuring surface to the measured object, and g is almost always g_n := 9.806 m/s^2 per ISO 80000-3:2006 (item 3-9.2) or similar standards. (Tbh the ISO document does not do much more than offer up a defined value.) This relation only holds (and only approximately) when the scale is used as intended by its designer, and that condition picks out a set of preferred (and accelerated) reference frames. A "contact camp" supporter (like Baez[1]) would say that the measurement of weight is accurate in a much wider variety of frames, crucially including in local free fall.
Modern weight-measuring scales will almost always use SI units internally, but most can supply a variety of output units (the one in my bathroom can show kg, stone, lb, Newtons, and the value read from the piezo element package. It sadly has no internal bubble level or equivalent, and no accelerometer, and its software doesn't make use of data from devices within the mobile device it runs on (which could be placed on the scale between my feet). Finally, it was expensive enough to do any or all of the above, so I won't recommend it).
More colloquially:
http://www.livestrong.com/article/343183-the-differences-bet...
[1] http://math.ucr.edu/home/baez/physics/General/Weight/whatIsW... -- and revisiting this wonderful page is the real reason for this reply. :-)