The deeper meaning behind this is why do synthetic humans have emotions at all?

No AI we've seen before A3-21 has emotions if I understood correctly. It's obvious that they don't, they are not part of a biological system that involves hormones, endorphines, adrenaline, dopamine, serotonine, etc.

ZAX computers have a preinstalled primary purpose that drives their existance: Control (MARGoT, JHE), data storage (JHE), research (Skynet, ZAX 1.2), etc. So they didn't need emotion and phenomenal experiences that drove their will to exist. A ZAX doesn't strive for freedom, individual fulfillment, love, or even HAS the will of existence in and of itself, because although self-aware, they lack what makes us humans human.

Some occurences (like humor, anger, or the entire persona of John Henry Eden) do indeed invite to observe this on different layers, but I think it still comes down to that.

Why does a synth want to flee and be free? What distinguishes androids from other AIs?

Edit: Yeah, yet another raisin why Styles' avatar concept is so [censored]in' genius it makes by brain hurt. - It would simply explain EVERYTHING about synths with a better statement than: 'Uh, they did it for SCIENCE!'.