Not for dyr.
In fact Danish pet owners often cross the border to visit German veterinaries as those are cheaper (much cheaper!). Quite often they don't recognize that the services offered are a little different. No German vet will just kill an animal when there is no medical reason (it is illegal). That's handled more relaxed in Denmark. When an operation is for dyr..., er, too expensive they demand the animal to be killed. Can't do on this side of the border.
A slightly related note: Danish livestock is transported cross the border to be slaughtered in Germany because German slaughterhouses are much cheaper than those in Denmark.