That video does look like it was recorded by a Tesla's built-in dashcam feature, but the claim that Autopilot recovered from the skid is almost certainly a lie:
That said, as a human driver I've managed to recover from incidents similar to this one (though none in my Tesla so far!). It's possible, but I don't recommend trying it. 😉
I've heard that Teslas are pretty great in snow and ice, but I haven't yet driven mine in those conditions. It does do a great job in the wet though. I've driven it in everything from apocalyptic Texas downpours to daily Portland drizzle and it always feels solid and planted.