My initial thoughts are that the fob is using low frequency (I think, don't quote me on that) RFID tag in the remote, as well as likely just a simple radio communication.
How close do you have to get to it when the power is off to start it? And when you say you can start it from far away, do you mean click a button on the remote to start it? Or have it be a few feet away and start it by clicking start on the car? My guess would be the first.
Traditionally, key fobs use radio transmission, where each unit is powered individually to work. This has a long range. This is preferred to UHF RFID because you need to click a button on your remote to unlock or start the car. With UHF a person could unlock your car in the driveway if the keys are on the counter (in some homes).
With low frequency RFID, the tag is powered by the reader when in close range, usually no farther than a few inches.
Assuming you have a smartphone, you've likely got a 3.something MHz RFID reader and writer in it. Download an RFID reader app, enable NFC, open the app, and touch the tag to the back of the phone. If you are looking for tags in the app, and the RFID tag is the same as your phone, you'll see a tag pop up. If it doesn't, it's likely a 100 something kilohertz tag, so then you know what frequency to buy.
Hope this helps!