Yes, you do need to remove what remains of that drain. The new drain is designed (like any other) to glue to the riser pipe.
Depending on how far down the trap is (i.e., the length of the riser), you may be able to use an internal pipe cutter to remove the old trap, part of the riser, then install a coupling, short stub and the new drain.
If the riser is not long enough, then you need to get more destructive, and may need to open the ceiling or tear up the floor to cut back the trap. Note, it is possible to ream out the trap socket with a fairly inexpensive special bit. Then, you could forget the coupler and insert a new short section as a riser and the drain assembly.
Note - it is EXTREMELY important to get the top of the drain level; more so than with a more conventional one (but even then, it is very important as opposed to extremely important). Because of the diameter of the Kerdi drain, any error means your tile around that area might have a hump if it is not set level. There is usually enough play when gluing up a PVC joint to make it work out, but sometimes you need to weigh it down while the solvent evaporates. It is also important to get the drain at the proper height. If you are using their foam tray, you can take the center ring out and use that as your height gauge. If you are doing a mortar bed, over the plywood, you want it to be in the order of at least 3/4-1" thick under the flange of the drain. You might get buy with slightly less. To get the Kerdi drain to be bonded to the mortar bed, it helps to mix a little bit of lime into the deck mud (or maybe a little thinset) to make it stick there. You want that small section of mud to be a little wetter and able to be smushed into the grid openings of the drain's flange to ensure 100% coverage and stick.
Make sure to watch the video that comes in the box with the drain and follow up at
www.johnbridge.com for some expert advice. I've installed a grand total of one, but have seen it done by the manufacturer's VP in a demo at a class, where I got a bunch of good practical experience and advice from experts.