They had to be euthanized due to several antisemitic laws that forbid Jews from keeping pets. The rationale was that the pets owned by Jews ate food that was an overall waste of resources. It was also forbidden to give away animals to be looked after.
Victor Klemperer, a Jewish diarist in Germany who barely survived the war thanks to the marriage to his Aryan wife, has a poignant section in which he has to euthanize his 11 year old cat Muschel because of the law. He and his wife Eva had used Muschel as a form of distraction and hope,
The tomcat's tail is our flag, we shall not strike it, we'll keep our heads above water, we'll pull the animal through, and at the victory celebrations, Muschel will get a schnitzel from Kamm's (the fanciest butcher here).
On the day of Muschel's euthanasia in May 1942, Klemperer sadly wrote:
Someone else may find it absurd or even amoral, when so many are suffering because of the fate of their relatives. But I see how [Muschel's fate] pulls Eva down. Muschel is pampered, for his last meal he got veal, as in peacetime- am I hard-hearted if I secretly note: one pound- when a pound and a quarter are the weekly rations for two people? Am I hard-hearted if I wish the moriturus were over? today Eva said: "The little animal plays, is happy and does not know it will die tomorrow." -Is there anyone among us who knows: Tomorrow will you die?
fyi, a bit more in this post