Did you watch the video with sound on? The car alerts before the car in front brakes.
What I heard was an alert go off after the first (red) car slightly veered off to the right, allowing sensors to detect the second (black) SUV's abrupt stop.
Are you sure you're viewing it with the sound in sync? There's no veering when I watch until well after the audible alarm. But if you look through the red car's tinted windows you do see the break lights of the cars in front - presumably it's that breaking action which the radar unit is picking up.
My take on this is the red car wished to switch lanes beforehand and was distracted by that, thus not reacting quickly enough to the breaking. The right indicator goes on and it slowly starts to move across just before impact (but not at a speed / timing which would indicate an effort to avoid impact)
I see the cars in front of that car braking, as well as those beside the Tesla, before the alarm.
No other car can do this actually. Only Tesla
I think Mercedes has a feature called PRE-SAFE Brake which does something like this.
I drive one (C350e), and indeed it monitors traffic and brakes when another car in front does too. Now the question is: "does it really matter that the car can see through the car in front"? Because if the car is self-driving it will keep such a distance that it should be able to brake & stop safely if something happens in front of it. Meaning: the Tesla should still be able to come to a safe stop even if it only detected the nearest car was braking - just because of a -say 2 second- distance between the cars.
The whole radar unit and lots of the intelligence in it comes prepackaged from Bosch. So yes, plenty of high-end cars do this.
I work on these units as a hardware engineer for a manufacturer. They are very clever bits of kit and a lot better than the other offerings on the market.
And to support your comment - yes, this isn't exclusive to Tesla by any means!