The Potters used to own a cat. While little Harry was a little rambunctious, the cat adored him, as did all members of the household. When his parents died, the cat imprinted on little Harry and accompanied him to Privet Drive. There, it proved to be not only loyal to Harry, it kept outsmarting the Dursleys. And soon, there were kittens, courtesy of one of the kneazles bred by Mrs Figg.
And so it came to be that wherever Harry went, he was accompanied by a few cats. Whenever he was upset, there was a purring bundle of fur to soothe his nerves. And he of course repaid the affection in full. Harry adored his cats and there were always headpats aplenty. Aunt Marge’s Ripper once tried to chase Harry, but feeling the numerous glinting predatory eyes on him, he turned tail.
Thus, Harry comes to Hogwarts accompanied by numerous cats and kittens, for even if he tried to keep them at Privet Drive, they would find a way to hide in his trunk.
Potter’s Pride becomes famous at Hogwarts, the preternaturally smart cats all doing their best to protect their human. Peter Pettigrew takes one look at the situation and decides that it’s not worth the risk, he flees during Harry’s first year at Hogwarts.
Mr Filch takes quite a liking to Harry as Mrs Norris takes one look at the boy and rubs herself against his legs. Harry, well used to such behaviour by then, knew exactly where to rub for maximum effect.
Any female wanting to date Harry needs to gain the approval of all of his cats, as several girls at Hogwarts found out. They also found out that feline vengeance can be quick and stealthy.