If you look at just a single catalog on solenoids from a company like Cole Hersee, you will get an idea how many different types are available and what voltages they work at. A 6-volt will work on 12-volt but not in a continuous duty cycle or it may burn a wire over time. Working a 12-volt solenoid or relay on 6-volt is not an ideal solution. Some may work and some not.
http://colehersee.com.au/switches/solenoids/
They don't have any 6-volt types there in that link but you can see all the different design types. Some have fine windings for continuous duty and some have heavy windings for intermittent rapid duty. There are different working voltage rating for the main contact terminals as well. There are still a few that make solenoids for 6-volt service but no where near what there used to be.