The post you replied to questioned whether they were considered invasive everywhere and asserted that they were on a global list of the 100 worst invasive species.

All as counter to the claim that Australia was unique in considering cats a nuisance.

It seems that assertion is true:

    38. Felis catus

    Felis catus was domesticated in the eastern Mediterranean c. 3000 years ago. Considering the extent to which cats are valued as pets, it is not surprising that they have since been translocated by humans to almost all parts of the world.

    Notable predators, cats threaten native birdlife and other fauna, especially on islands where native species have evolved in relative isolation from predators.

    Common Names: cat, domestic cat, feral cat, Hauskatze, house cat, poti, pusiniveikau

GLOBAL INVASIVE SPECIES DATABASE

http://www.iucngisd.org/gisd/100_worst.php

Is there nowhere in Europe where cat populations are kept in check to protect birdlife and other local native species?

In Western Europe, not really to that end other than quite localised.

But even then the discussion has to be what is a native species (with birds that can get quite odd with species disappearing and reappearing hundreds of years later, for example). There is just no real nature left, i.e., more about deciding when to make a cut.