Hi ydukes,
the idea is that your vet first reads the data on the microchip of your dog. This is to make sure the ID on the dog's microchip matches the one mentioned in the pet passport and also to make sure the microchip is still in place and can be found if the customs officials choose to check the dog.
Then after checking the microchip, the vet gives the vaccination and finally fills the information in the passport. You do not fill in this information on your own.
If you like, you can fill in the pet's name and address and your own contact information. In my dog's passport there is even a place for a photo

but none of that are considered relevant info from the customs' point of view. What they care about is the info about the vaccinations and it is always filled by the vet.
I will be driving through Europe with my dog in June and just took her to the vet to get her shots and to update her passport. Now I just need to figure out how to transport her insulin from Finland to France. She is diabetic and the insulin needs to be stored in the fridge before taken in use. We used to have a mini-fridge for the car but it takes quite a lot of space considering I just need it for a few small syringes.
Happy traveling
--Penni