(Airbus via SKYbrary)
Is it true that for a given specific aircraft there is a single specific TAS that aircraft can travel on given FL (due to wings lift forces produced, etc)?
Above shows a common climb profile, and is the best illustration to the main title question. As you can see, what the plane would typically be programmed to target is IAS or Mach.
Related to that point and graph:
I thought aircraft are capable of traveling with a range of speeds at given altitude depending on Angle of Attack (AOA).
There are ranges of speeds, but the economic range (see: What is Cost Index?) is not that big (approx. ±20 knots), and the dataset you have simply shows the typical value ATM can expect, for simulation purposes for example as the website says:
The BADA APM is designed for simulation and prediction of aircraft trajectories for purposes of ATM research and operations. Aircraft performance parameters and trajectories can be calculated using information and data contained in BADA.
The filed flight plan would also include the planned TAS for the different flight levels, and as long as the pilots don't deviate by more than ±5%, they don't need to notify ATC.
Or maybe what I am looking at (...) the only one FL+TAS pairs that makes sense in practice (as flying otherwise would be highly inefficient).
In essence, yes. Also worth pointing out is that a lightly loaded jet-liner would typically aim to fly higher if permitted and if possible. The same is true even for short-haul flights (What should be the minimum time spent in cruise (for e.g. a B737)?). So basically optimization is solved by aiming for the optimum FL.