GM has created a confusing mess of terminology and branding on this topic.
Automatic 4wd is not the same as AWD.
I have the NP246 “Autotrac” transfer case with auto four wheel drive. But it doesn’t use “Auto-trak” transfer case fluid, it specifies Dexron 6 in my manual. I have seen other contradictory information elsewhere.
2008 Tahoe manual (Auto 4x4 the only option):
The AWD TC with full time AWD and the viscous coupler is a different item.
YMMV I’m sure they changed the spec over time.
For the OP; just check your manual, which is available online from GM if you don’t have the original.