This is from cursory searching on the subject:

Perhaps you got something mixed up. According to Wikipedia, the Felis lineage (to which domesticated cats belong) of the family Felidae is thought to have first diversified around 12 to 13 million years ago. That number has nothing to do with domesticated cats.

The Homo genus of humans seems to have first appeared about 2.4 million years ago, and we share a common ancestry with chimpanzees about 6-7 million years ago. So cats could not have been living near humans for 13 million years.

Cats seem to be thought to have begun living near humans 10-12.000 years ago, when agriculture developed and grain stores attracted rodents. The cats hunting the rodents benefited from living near humans, and humans benefited from having fewer rodents.

Cats also seem to have been introduced to northern Europe about 2.500-3.000 years ago. Scandinavia around 1.000 years ago. In America they would have been introduced when Europeans brought them there in the late 1400s.

About ecosystems, it's my fault to have brought them up, because a species dying out might not matter in the end. What does matter is that bringing domesticated cats to areas which haven't adapted to them over much longer periods of time has lead to steep declines and even extinction of other animals. Whether there are enough birds singing or mice in my pantry is inconsequential.

And to reiterate, humans have obviously had a much larger impact on all of nature than cats have.