Does it mean the same percentage of takeoff thrust might provide different thrusts according to the operator?
Not just per operator, it is different per engine installation in one fleet. That's why calculations are done according to each airframe's AFM. And as the engine ages, its thrust decreases* (and fuel consumption increases), and this is reflected in the calculations. (The fuel flow (and drag) offsets are based on data from previous flights, and are presented to the pilots as corrections/biases to input.)
The selection will finally show as a percentage.
The final selection will be percent N1, EPR, or for the latest planes (A350 and 787 with R-R engines) as an actual available thrust percentage. So it depends on the plane/engine/technology.
As to why, because lbf or kN mean nothing at first glance.
- The calculations are very complex to do by hand.
- Density altitude would make a correct number feel incorrect. In other words, 95% N1 (for example) is a different thrust force at SL at 15°C from at 5,000' MSL at 35°C.
The number simply shows without guesswork or confusion** how much of the available thrust will be used. And it makes the AFM (or EFB) easier to present/use.
* As an example, a heavily used fan will not be in a good condition, so two similar fans (but different ages) running at the same rotation (N1 percentage) won't deliver the same force. (The same for the conditions of the rest of the stages and combustor.)
** Plus, the engine controllers have no way of measuring force, this is another reason why the selected level during flight prep must match the engine instrumentations, otherwise the crew can't verify the takeoff thrust that's been preset, has been achieved.