Is it possible for a human to get rabies from a rabies vaccine meant for dogs?
It would depend on which vaccine you're talking about, as there are several. Some use attenuated strains of the Rabies virus, others use killed, and some are recombinant glycoprotein vaccines. Here is a list from the CDC of all the rabies vaccines available in the US in 2011 (I wasn't able to find anything newer). Generally speaking, through, one would expect the FDA in the US and the relevant authorities in other companies to require vaccine manufacturers show that their product is safe in other species, in case of accidental exposure, such as you are talking about here. This is especially true of something being given to an animal that is quite often a family pet.

If you think you've been exposed to any type of medication which has not been prescribed for you, including dog rabies vaccine, you should seek medical attention immediately, if not sooner, but based on my experience in the pharmaceutical industry and dealing with regulatory agencies, you should be pretty safe, as the regulations surrounding vaccines are quite stringent.