No you are not mistaken. That green wire is the neutral grounding wire. They tell you to attach it to the neutral, since that is where it originated from, and they don't want any loose bare wire ends floating about.
Manufacturers have begun using the wire pigtail as a bonding jumper in lieu of the older bonding strap these days. The problem arises when someone had a 4-wire cord installed, and then later moves the appliance to a new premises which only has a 3-wire outlet available.
Permanently removing the bonding strap leads to such appliances being reconnected with a 3-wire cord, and then operating without any bonding or grounding connection.
Just today, I was asked by a customer to install a new dryer outlet, since they recently moved into a new house which only had a gas dryer connection available. The dryer is several years old, with a 3-wire cord attached.
I installed a 4-wire circuit and NEMA 14-30R receptacle as required. I also changed the cord to match. While in this process, I noticed some potential hazards, so I took some photos to document what I found. The date/time stamps are accurate.
You can call the first photo "Before" -- this is how I found the appliance. No connector, and screw missing from access panel cover.
The 2nd photo is "before" with access cover removed. Notice that one of the hot wires has begun to melt, due to a loose connection. Also note that the bonding strap is missing.
3rd photo shows the new cord installed, with all connections tight.
Last photo shows the instructions on how to install a 3-wire cord on this appliance. It is a sticker on the back of the dryer.
Now the question comes to mind: What if they had moved into a house with a 3-wire outlet? They would never had called for an electrician, and simply plugged in their defectively wired dryer and went on with life.
That is until the loose connection melted the cord sufficiently to short out the hot line to the frame of the appliance. But, with the bonding jumper missing, a definite shock hazard would have manifested itself between the grounded washer adjacent, and the electrically hot frame of the dryer sitting next to it.
How would the lady of the house been able to deal with being zapped while transferring wet laundry from the washer to the dryer under such a scenario?