Military pilots/crews that fly airshow demos are specially picked (often a political process), carefully trained, and the maneuvers precisely prescribed.
No idea if this crew was in that category... Was this a demo? Or just a S.H. takeoff?
Airliners are built to climb with an engine out, at max gross weight. If you're really light, and all the engines are at max thrust, you can get quite a bit of excess thrust. This crew climbed at what looked like a crazy angle, but if the airplane was accelerated a bit longer on the runway, used a low (less drag) flap setting, and carried a bit of extra speed into the rotation and climb, it could've easily maintained airspeed at about 30-40* degrees nose up with max thrust at low weight.
And that's about what this looked like. Very steep for a transport category jet, not usual, but within the performance of the airplane.
You have to maintain an airspeed that allows you to full control if an engine quits in that climb. The speed known as V2 is the minimum at which that's true. I suspect they were quite a bit above V2. V2 is often quite low, like 140 Knots, even at max thrust.
*In a lightly loaded 747-400 one day, taking off out of Denver, I had to maintain 25 degrees nose up to keep from exceeding the flap speed. Full passenger load of about 400 (so, 80,000# of people, plus about 20,000# of their luggage), but only 100,000# of fuel. So, about a 600,000# gross weight, in a jet that was usually at 875,000#.
We were at reduced thrust, at high altitude, but we were light. Take that same jet, without passengers, put it at sea level, and use full thrust, and yeah, I could see a climb like the one in the video.