Whilst the adafruit displays use a variety of LCD driver chips (ILI9340/ILI9341/ST7735R/HXD8357D), all of the red TFT LCD boards I have enocuntered so far all work with the Adafruit_ILI9341 library, so I'd presume they are all using ILI9341 driver chips. This is for the 2.2" LCD I got with the Oak, as well as some 1.8" and 2.8" displays I bought on ebay. I am using the Oak TFT shield, but I didn't solder the display directly on, only headers. I then use an extension to link to the display. I can swap out any of the three displays when using the graphicstest_esp8266 code without any changes, and all three sizes work flawlessly (just change the display when the Oak is not powered!).
Interestingly enough, in a discussion about which library to use to control the displays, Paul Stoffregen (one of the people contributing to Arduino) commented that at least on the older Adafruit display with ILI9340, the same code work with either [ILI9340 or ILI9341] controller chip.
I am using the 1.0.5 Arduino IDE Oak board package, Arduino 1.6.10, and sketchbook that has only Oak related code and a couple of custom libraries. There is no other copy of the Adafruit_ILI9341 library installed in this sketchbook, so there is no change it is using a different version (thus also no warning in the compile log about the IDE auto-magically picking between multiple libraries - and getting it wrong). And the example is working just fine in my case.
If you are thinking about trying the Adafruit_ILI9340 library... you may not need to do much. Both the Adafruit_ILI9340 and Adafruit_ILI9341 libraries have been updated with ESP8266 support (although I haven't had any need to test that out myself yet) since Erik starting working on the Oak, so may actually be fully compatible already - at most they might need to be tricked into thinking they're being compiled for an Oak instead of a ESP8266.