The bare copper wire and any green insulated wires are meant to be connected together, making it a "party of three" as you stated in your question. That said there is another consideration to take into account as well. These safety ground wires are also meant to be connected to a common ground point of the electrical box as well.
If one of these wires (green or bare copper) coming out of your box already attaches to a safety ground point in the box then you have no more to do than the "party of three" connections.
If neither of these two wires are connected to the electrical box safety ground then it will be necessary to add an additional pigtail green or bare copper wire to the box safety ground and join it with the other three making a "party of four" connection.
In a perfect world, with perfect electronics, there would be no need for a grounding conductor. It's there to deal with imperfections. The most likely case for a ground fault in this installation would be if the black conductor lost some insulation and shorted itself against the frame of the fan. In such a situation, the casing could become electrified, and a grounded person, touching the fan body could receive a shock.
If it were properly grounded, you'd get a short circuit to the ground wire, and the breaker would trip.
However, the fan is mounted to the ceiling, and the possibility of someone touching it is low. (Unlike, for example, the frame of a fridge or stove, where human contact is part of normal operation.)
So, Is it safe? Quite likely. For added safety, turn the wall switch off when changing bulbs. Is it to code? No. -- Unless, as others have suggested in comments, the mounting box is grounded on the other side.
You can however, locate the breaker, and replace it with a Ground Fault Interrupt breaker, which would trip should a ground fault occur. This isn't something you should do yourself unless you have experience working in the panel.
Another possibility, if your switch happens to be chained off of an outlet, which is quite possible, depending on the routing of the wiring on that circuit, is to replace the outlet with a GFCI outlet, and wire the switch to the load side of it.
I am installing a Z-wave switch (I have done others), but this one is different. There are three switches in this junction box. However, only one switch is grounded (not one I am installing), and the switch I am adding needs to be grounded.
Best Answer
Looks like the left switch is grounded, the middle switch is not grounded but should be and the right switch may be grounded (can't tell from the pictures). Since the box is plastic, there is no way that they are all grounded. Run a wire to pigtail the middle switch (and possibly right switch) to the ground wire.