This is called sequencing. The idea is to distribute the high load on the hydraulic system due to retraction over time, so the hydraulic pumps can be smaller.
The linked video shows no such sequencing. The triple redundant hydraulic systems on airliners do not need such optimizations since there is plenty of power if all pumps are running. On top, the video was made in a hangar and the pressure source was most likely external, with enough power reserves. Any asymmetries are most likely due to differences in friction, as Jan remarks in his answer.
Since friction losses in the hydraulic lines increase with flow speed, a difference in retraction speed would quickly be equalized, since the faster moving leg needs higher flow speed in the lines leading to and from its actuator, and less pressure would be available for gear operation. Active syncing is not needed, and a small difference has little influence on flight characteristics. After all, the main gear is close to the center of gravity, so drag differences between both legs do not produce big aerodynamic moments.
When a gear retracts, it first needs to move out of the locks, which requires little hydraulic power. After this, the load gradually increases the more the leg moves up into the horizontal position. Shortly before moving into the final, retracted position the load is highest; the details depend on the gearing of the retraction mechanism. If this load needs to be covered for all three gears at the same time, the hydraulic pump on an aircraft with manual controls and hydraulics only for gear operation would need to be more than twice as large as when you sequence the retraction, so the peak loads come one after the other.
Rockwell 112 while gear retracts (picture source). Some asymmetry is clearly visible.
Technically this is solved with sequencing valves which serve several circuits. They are used both to time the gear door operation, so the doors open before the gear starts moving, as well as the sequence of the main gears.