Upvote:4
The process is reasonably simple: you just take your pet and EU health certificate, the one you have endorsed by APHIS, to a veterinarian in the EU. Most vets can issue EU pet passports; just call and ask to be sure when you make your appointment. You have up to four months to do this, but if you plan to travel with your pet around the EU you should do it as soon as possible, as airlines and train operators will ask for the pet passport even for travel within the EU.
It is indeed possible that your pet might have to be vaccinated for rabies again.
A writer at The Points Guy recently moved a pet to the EU and documented the process. You may find it interesting additional reading.