I dunno why you were flagged. It's a totally valid question IMO, and I asked it a lot in school too (environmental science undergrad). To me, the difference is the RATE of change and expansion.
Nature and evolution can pretty easily deal with small changes in small areas happening over hundreds or thousands of years, which was normal before human population explosions.
But when that change happens across multiple human settlements (large areas), quickly (industrialization and the green revolution in just a few decades), and at a huge scale (not just cats but climate change and deforestation and pollution all leading to habitat loss), it is very very hard for other species to cope.
If a few individual wildcats crossed a land bridge, sure, they'll have a field day as they gorge on unsuspecting prey. They'll do well and reproduce, and their kittens will eat more prey. But eventually there are more wildcats than prey, a harsh winter or drought comes along, and a bunch of kittens die and the prey have a chance to recover over the next few years. It reaches a dynamic equilibrium.
That's different than when humans settle a new place and suddenly introduce hordes of rats and start breeding pet cats, at numbers far above what the prey population can sustain (because we feed them pet food). The unwanted ones become feral and enter the system at a rate higher than they would've through normal migration, and then we keep breeding more and more of them with an outside input (more pet food). Pretty quickly the feral cats will kill off a lot of the unsuspecting prey (the dumber, more exotic birds, often) and leave a bunch of human-adopted species like rats and corvids and squirrels that can coexist with cats, often having immigrated with the humans for hundreds of years. So you end up swapping the local wildlife for the introduced ones in just a decade or two, way faster than the dozens or hundreds of years it usually takes.
On islands or where settlements and road networks form unnatural barriers, this is especially bad because the escaping prey have nowhere to run to, being isolated to smaller non-contiguos pockets that can individually be trapped until the whole species is extinct or at least extricated (locally extinct).
Of course there are gray areas. The Hawaiian Islands, for example, are a mix of species that ended up there naturally, some a long time ago, some more recently. They have different times of introduction, just like humans do. But because it's a young island chain (in geologic time), its whole timeline is compressed and the species there have different, eh, degrees of "native"-ness, I guess. And among the various introduced species, not all of them are considered invasive. https://dlnr.hawaii.gov/hisc/info/
Anyway, it's not an exactly cutoff date or algorithmic formula for determining nativeness. It's just a case by case judgment call based on the relative time of introduction compared to the ecosystem age, and also the effects of the introduced species on other preexisting ones. It's a human judgment call subject to errors, like any others, but still a useful consideration in ecology.
Sorry for the long post...