No, we don't. The only requirement is a chip + rabies shot if you are taking your cat, dog or ferret with you across other European countries (per directives 998/2003 resp. 576/2013).

> My ex lives there, has lost her cat, it was captured by animal control and she had to pay a 300€ fine at the animal shelter to get it back.

That is not a fine by the government, it is a charge from the shelter for the trap, medical checkout and accomodation cost.