Everyone has their own idea of what goes where, much depends on your local city policies on recycling.
For me, I use the (almost clear) applicator lid as an ant trap for my hummingbird feeders. I drill a small hole in the center of the lid (through the center post without damaging it) and slide a metal S hook through... and then fill the lid with water.
As for the rest of the device, I don't think there is any more DNA on any part than a Kleenex might have when blowing your nose and the sensor that goes under your skin is not a needle, so it doesn't need to go into your bio-hazard or sharps container. Though it does easily pry out of the sensor if your really really adamant put that in the sharps and then the rest of the sensor which I find hard to break open, contains a circuit board and battery. So that part really should go into your battery recycle. I just put the whole sensor into the battery recycle.
And as for the applicator, which has short needle to get the sensor filament under the skin, it is so protected that you'd have to break open the applicator to get at it, can just go in the regular trash. And again, any DNA on that needle is so small of an amount... I don't worry about it.