Thank you for articulating what was bothering me about this.
I couldn't quite put my finger on it, but you're right. They are confusing defining the system with defining the entropy of a system and then saying it's the entropy that is subjective. That isn't the case at all. Entropy is just a measurement.
That can't be right, because the arrow of time depends on the universe being in a high entropy state when the Big Bang happened. There were no observers then. Also, the 2nd law of thermodynamics doesn't depend on observers. The universe overall will continue to evolve into a higher entropy state without any intelligent observers making measurements. As in stars will burn out and blackholes will evaporate, and the temperature of the universe will get close to absolute zero.
> the 2nd law of thermodynamics doesn't depend on observers
Thermodynamics entropy is not just a property of a system - it depends on how we choose to describe the system. Of course, we can define the thermodynamic entropy for a system without observers. (In fact, we can only define the classic equilibrium thermodynamic entropy for a system without observers!)
I couldn't quite put my finger on it, but you're right. They are confusing defining the system with defining the entropy of a system and then saying it's the entropy that is subjective. That isn't the case at all. Entropy is just a measurement.