Crookshanks: Harry's first pet?

In the deathly hallows, it says at some point that Harry had a cat when he was a child, before his parents died. Is it possible that Crookshanks could have been that cat? In all of the books after Crookshank's appearance, he always took to Harry, and took to Sirius when he was trying to get to Wormtail, perhaps a sign that he already knew Sirius? or am i just looking way too much into this...