I've really never used the Arduino IDE, so I can't comment on that, but I don't think you're going to be able to use standard Arduino sketches that use the SD card with the Edison. You are correct that it is going to use the SPI/SD interface directly, and there's a real SD card driver (and filesystem driver) on the Edison.
When you plug an SD card into the Edison it mounts it just like any other drive and you can access files just as you can normally in Linux (or Windows for that matter).
In C you would use fopen/fread/fwrite/fclose to read/write files. In C++ you can also use those or iostreams. I don't know for sure that those will work with the Arduino interface, but I think they should.