Are you sure the suburban axle has the same gear ratio as your original yukon axle? If the rear diff ratio doesnt match what the PCM and ABS module are programmed for, it will think the wheels are spinning/losing traction, and then activate traction control.
And, your yukon doesnt have "stability control", it has "traction control" only. The difference is that true stability control will actively brake each wheel individually to correct yaw/oversteer-understeer and help prevent rollovers under hard steering at higher speeds.
The traction control only system that your yukon is pretty basic, really it just prevents/reduces wheel spin under hard straight-line acceleration and thats about it. It can ask the PCM for a torque reduction to stop the wheels from spinning, but it cant actively brake the wheels by itself and it doesnt have a yaw sensor or steering wheel position sensor.
"traction control" was a 2wd-only thing starting in 2000. 4x4 SUV's had nothing in the form of traction/stability control until 2002.
"stability control" was not available on any GM fullsize SUV until 2002, and in 2002 the ONLY vehicle it was available on was the Escalade.
Then in 2003, true 4-channel stability control became standard on the Denali (and escalade), but it was not available on the tahoes/yukons/suburbans.
In 2004, stability control became optional on the "non-luxury" SUV's (tahoe, suburban, yukon)...but it was a pretty rare option and could only be had with the unique AWD transfer case. It was not available on the tahoes/yukons/suburbans with the standard auto4wd/2hi/4hi/4lo transfer case.
Then in 2005, the oddball AWD/4low transfer case went away, and you could get stability control on the tahoes/yukons/suburbans with the regular auto4wd/2hi/4hi/4lo transfer case...but stability control was still optional. Much more common option though in 2005, most fully-loaded SLT/LT tahoes/suburbans/yukons had it as part of the top-trim-level package.
Finally in 2006, stability control became standard on all fullsize SUV's.....due to the increasing number of rollover accidents/lawsuits/etc against GM.
2wd SUV's didnt get true stability control until the 2005 model year if I remember correctly...
but basically to answer your question. I think you have the wrong rear axle ratio. You'll have to either swap gears, or have the PCM and ABS module reprogrammed for the different ratio rear axle.
Look in the glovebox of your yukon, and of your friends suburban. Look for an RPO code that says "GT4" or "GT3" or "GT5". Thats your rear axle ratio (it will be one of those aformentioned codes)....then look in the glovebox of the suburban, and see if its different.
ben