OP already discovered the answer in that there is a Power Transfer Unit( t-case )and Rear Drive Module( rear diff ), in addition to the trans, that need fluid replacement as the Equinox is an AWD. Couple things...
1 - I had an Equinox( 08 AWD ). There is no trans filter to replace on the Equinox's equipped with the 5 spd Aisin trans used behind the 3.4L unless the whole trans is removed and taken apart. All you do is pull a drain plug( on the right rear as I recall )to drain it. Takes approx. 4qts to top off fluid again( no normal dipstick to check fluid level - it is hidden behind a nut somewhere ). Filter is not a normal service item on those Aisin transmissions. Many Equinox owner's were adding inline filters that can be changed when I had mine. Fluid to use is T-IV( 08+ sport models with the 6 spd trans used Dev VI just FYI ). Any multi purpose ATF that lists T-IV will do the job.
2 - The PTU's and RDM's used different fluid based on model years and even which engine/trans the vehicle had( 08+ ). The early models, like the OP's 2006, used a special GM fluid called Versatrak in both the PTU and RDM while later model years like my 08 used 75W-90 synthetic gear oil( GM's Grape Juice )in the RDM and Versatrak in the PTU( Sport w/ 3.6L used 75W90 ). It really flipped flopped around. You need to make sure which is used in your specific model year. For the OP with a 2006 it is Versatrak in both the PTU and RDM.
3 - GM's Versatrak fluid is another one of those "special" GM fluids that does not have a lot of aftermarket options. IMO use the GM stuff. However, there has been some success using RP's Synchromax fluid as a replacement. RP lists it as a replacement and lots of people have used it with good results while others have not faired so well and reported grinding. The Saturn Vue crowd seemed to have the best results replacing it with the RP stuff. There was a member here that had an AWD GM minivan type vehicle that tried it and it did not work for him though. So, it may or may not work for the OP in his Equinox? That is why IMO it is best to just stick with the GM stuff.