Could the Boeing 787-9 near vertical takeoff demonstration be performed by Airbus A350 aircraft?
No.

Firstly, as a comment noted, the takeoff was hardly "near vertical", the camera angle makes it look so. The takeoff angle in the video certainly is much steeper than a normal one, but Airbus aircraft are more than capable of doing the same - in fact an A350 did very much the same thing at the Paris Air Show. 

The notion that an Airbus doesn't let the pilots have complete control is a little bit misleading. I'm not going to go into too much detail in this question, but what most people are referring to when they say that is the Alpha Protection system. This system kicks in when the aircraft is at extreme angles of attack. For example, if an aircraft is about to stall, it will not let the pilot pull the nose up. In any plausible situation the pilot would not want to raise the nose anyway, yet we have seen accidents such as Colgan 3407 where the pilot did exactly that (and, also AF447, however the system was disabled in that crash). 

In summary, the Airbus system pretty much only prevents pilot actions that are dangerous (OK it is more complex than that, and the Boeing design has benefits too, but..). Back to your question, why does the Airbus system also allow such a steep takeoff? Because only the attitude was high, but the angle of attack was still within normal limits. These takeoffs are possible on lightly loaded aircraft with engines near full power.