Are thrust levers synchronized by default when pushed/pulled?
On a non-FADEC engine (older airplanes - not this one) where the thrust levers are connected to the engine fuel control units with a cable circuit or a teleflex cable, you have to manually adjust each lever to wherever it has to go to achieve a given N1 (fan speed), and this can result in small variations in position of each lever when all engines are at the same power setting.  On takeoff on a non-FADEC, you move the thrust levers until power is reasonably close to take off thrust, call "set thrust", and the non-flying pilot tweaks each lever to synchronize all 2 or 4 N1 values to the takeoff bug setting on the N1 indicators.  With mechanical controls, the thrust levers may or may not be perfectly aligned when the engines themselves are in synch (there are limits to the misalignment allowed).

On a FADEC aircraft like the 380 in the video, the engines are managed by computers, and the thrust levers are just electrical input devices.  At the bottom of each lever there are Rotary Variable Displacement Transducers (RVDTs), that tell each engine's FADEC computer where the thrust lever is.  The FADEC will manage the N1 and if all 4 levers are at the same position, each FADEC will achieve the same N1 (within the tolerance of the RVDT's signal).  

For takeoff, to make it easy, there is a detent for each lever at the Takeoff Go-Around setting (TOGA) that you can feel as you move them up (a little roller drops into a depression on a cam profile on the lever's arc of rotation).  There is another detent for climb power as well (CL) to avoid having to fuss with lever settings during the departure. When in the TOGA or CLIMB detent, the FADEC computer takes care of setting its engine's computed takeoff or climb power automatically.  So on a FADEC engine, when you take off, you just push the levers until you feel each one click into its TOGA detent, call "set thrust", and the non flying pilot just looks at the N1 indications to confirm they are all at TOGA thrust, without having to tweak the thrust levers (on the RJ I used to touch the levers below the handles as the PF was pushing them up with my finger and thumb to feel for myself that the levers clicked into the detents, although it wasn't really necessary).

At settings other than TOGA or CLIMB thrust however, they tend to work like mechanical thrust levers, where if one lever is at a different position it will have a different N1 value, so at intermediate power settings or on approach you have to individually align the levers, so you end up moving them as together as best you can and fine tuning as required.

What the pilot in the video is doing is synching the levers for taxi because when he pushed the levers up he didn't push them evenly and it looks like the outer ones were a little farther ahead than the inner ones.  Not that big a deal for taxi; he's just being a perfectionist.