What is the ruling of Islamic law regarding preventing a female cat in the house from mating with its kind, and is its owner sinful for doing that?

There is no specific legal evidence that necessitates observing a particular aspect for someone who acquires and confines an animal. The legal proofs indicate that a person's responsibility is absolved if they feed it, as in the hadith of the woman who entered Hellfire because of a cat she tied up and did not feed. This is also evident from the hadith of Al-Nughayr regarding Abu Umayr, from which Ibn Hajar inferred the permissibility of keeping a bird in a cage and clipping its wings.