It's highly unlikely that Crookshanks was the Potters' cat.
If Crookshanks was the Potters' cat, he would have then been around 13 years old when Hermione got him in her third year. In addition, Crookshanks is around after that, and shows no sign of being an old or aging cat. Scabbers was only able to have such a long lifespan since, being an Animagus, his was a human lifespan and not a rat's. Crookshanks is part-Kneazle, but Kneazles aren't mentioned as having extraordinarily long lifespans either.
When the sales lady tells Hermione that Crookshanks has been there for ages, it likely doesn't mean years. People say things take "ages" metaphorically sometimes, like "this food is taking ages to come, the chef is so slow". So, it's more likely she meant Crookshanks has been there longer than most pets, rather than actually for years.
In addition, Crookshanks didn’t seem to recognize Sirius like he probably would have if he had been the Potters’ cat - he realized Sirius isn’t a real dog, but he takes a while assessing him before trusting him. Sirius seems to be new to him rather than a familiar figure he was now unsure of.
“This cat isn’t mad,’ said Black hoarsely. He reached out a bony hand and stroked Crookshanks’s fluffy head. ‘He’s the most intelligent of his kind I’ve ever met. He recognised Peter for what he was straight away. And when he met me, he knew I was no dog. It was a while before he trusted me. Finally, I managed to communicate to him what I was after, and he’s been helping me …’
‘What do you mean?’ breathed Hermione.
‘He tried to bring Peter to me, but couldn’t … so he stole the passwords into Gryffindor Tower for me … As I understand it, he took them from a boy’s bedside table …”
- Harry Potter and the Prisoner of Azkaban, Chapter 17 (Cat, Rat and Dog)
As for why Crookshanks chased around Scabbers, it's because as part-Kneazle, he'd be able to detect suspicious and shady people (or rats) easily.
“The Kneazle has an uncanny ability to detect unsavoury or suspicious characters and can be relied upon to guide its owner safely home if they are lost.”
- Fantastic Beasts and Where to Find Them
Crookshanks did indeed realize Scabbers was trouble - but that's likely because of his Kneazle senses and not previous experience.