I noticed this when I first saw it. I’m always surprised how surprised other people are when I point this out. The themes are there throughout the franchise but they are more explicit and complete in TS3.
Toys are sentient but aren’t allowed to be
The gimmick of the movie is that toys are alive. They have emotions and dreams and can make independent decisions, but the other side of the gimmick coin is that it is their purpose to act as if they don’t.
Children are God
Toys are expected (for some reason) to regard their owners as infallible. The moral of the story is that the toys are supposed to have faith in Andy, their god, regardless of anything that happens in reality. It is, again, their sole purpose to serve him.
Secular society leads to tyranny
The toys end up in a “community” in which toys have no single owner. Religious fanatics often equate aspect of secular society with polytheism, arguing that the one true god has been replaced with myriad false idols. The consequence of this is, of course, a tyrannical state system in which toys are kept in boxes and eventually destroyed by uncaring false idols . Once they’re broken they even go to a hellish furnace.
 This also resonates with Jobsian philosophy / economics. The idea that things that are owned are looked after better.