You will want to examine the device (in the photo it's called a "Scentee") and have a good look at the 3.5mm plug. I suspect its plug may look different from a regular audio plug. Even the "ordinary" earbuds that come with an iPhone use a plug that has more metal "rings" than a standard 3.5mm stereo audio cable. I believe that's because the iPhone earbuds include a microphone. The iPhone's 3.5mm connector probably has some special contacts in it that touch the different metal rings to detect what sort of cable you have plugged into it, and the software may not even send the special signals when a regular audio cable is inserted.
Have a look at How does Square Reader (squareup) device work via earphone jack in Android? and What protocol does a headphone jack use? -- these topics address different scenarios, but I think you might be able to glean some technical details from them.
I suspect you may be able to connect your Scentee device to your Pi, but depending on its plug, you may need to cobble together a special 3.5mm socket and some additional wires and connectors to get the input and output signals to the right places.
As for rewriting the SDK, remember that iOS software tends to be programmed in an Apple-specific software environment, so you may need to make lots of adjustments to fit any of the open-source software environments available on the Pi.