Goal tolerance and trajectory repeteability with MoveIt on real robots

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On the first question, no, with the default planners there really is no way to get the exact same trajectory every time you run things (especially on a real robot). This is mainly due to two things:

• MoveIt primarily uses OMPL. The planners in OMPL are sampling-based, thus you are almost guaranteed to get different samples for every plan even if the start and end pose are the same. If you want highly repeatable trajectories you really need to use something like SBPL - but I don't think there is a maintained version of arm planning with SBPL these days (it can be quite a bit slower, so mostly people stick with the unpredictability of OMPL).
• On real hardware, your "start" is never actually the same. Even if the arm looks to be in the same position to you as a human, that position is represented by a floating point value and there is going to be small differences in the floating point positions.

As for the second question, yes, this is almost certainly due to the robot drivers. MoveIt just makes the trajectory, how well you follow the trajectory is up to the drivers and hardware. Depending on the tuning of the drivers/hardware, different trajectories will likely end up with different accuracy in getting to the goal.

A final thought is: how are you determining that the trajectory is off by 2mm?

• Are you using TF tools to measure how far off the robot thinks it is? If so, then this sounds like either the drivers/hardware has some tolerance set higher than you like.
• Are you measuring that difference in the real world? This is likely more a factor of "global accuracy" - while most industrial robot arms specify very tight repeatability they often have much wider tolerances for global accuracy. Since most industrial robot arms are programmed by moving the robot to a specific location and then recalling that location, repeatability is what matters to them (if you put the arm joints at the same value, the end effector ends up at the same place every time). Global accuracy is quite a bit harder (small errors in the kinematic model due to part tolerances and stackup cause relatively large errors in the global accuracy).