For years my neighbor has been acquiring cats. The cats breed and often the kittens run in the building. Does my neighbor have ownership on these cats or am I allowed to remove them from the area since they are a pest.
The cats are probably your neighbor’s possession, since in principle they are hefker, and your neighbor has acquired them.
However, the building is private property, and it is forbidden for any neighbor to make use of the property in a way that disturbs other neighbors and is not a common and accepted way of using the property. The breeding of cats and allowing them to roam freely around is not accepted use, and therefore you are certainly within your rights to inform the neighbor that his use of the property is unacceptable, and that he should desist from it.
If he doesn’t stop, you can also get rid of the cats.