If I had to guess? It's probably something CyberLife programmed into him. So, probably closer to your second theory.
We tend to see Connor doing coin tricks in downtime—while he's riding up elevators to investigate Daniel and Stratford Tower, when he's walking between bars looking for Hank, when you idle him for too long. In all likelihood, an android that shoots with sniper precision, performs high-speed parkour stunts, and needs to be able to reconstruct/preconstruct using angles and trajectories and such requires immense precision. In all likelihood, he's calibrating, making sure those systems are in perfect working order at all times.
As for why a coin trick vs. something else? Marketing. Canon holds that there's a humanization department at CyberLife that codes in subtle fidgets, idle behaviors, blinking and breathing and fidgeting and twitching into the androids. But Connor's not just a run-of-the-mill household android that you don't want being all uncanny valley while it's idling between chores. Connor is billed as a sophisticated, high-end, costs-a-small-fortune prototype for a new generation of high-performing investigative units. So how do you convey through idle motions that this machine is precise, skilled, able to perform with far greater precision than a human? You have him show off cool party tricks that take concentration and dexterity without even looking directly at the coin. Like "oh, of course we can trust him to snipe things if he can catch a coin on its edge out of the corner of his eye."
Some bigwigs at the humanization department probably got together with the engineers and had a brainstorming session to figure out the most obvious demonstration of casual physical superiority they could have him perform while also calibrating his systems. And then Some Guy In the Room was like, "hey, my cousin did some cool coin tricks at the last family reunion, that would be pretty nifty."
I think if fidgeting like that was actually important to Connor himself, or if it was a soothing/stimming thing, he'd be doing it in emotionally high-stakes situations (i.e. going back to meet up with Hank in the good ending, or while he's waiting for Markus to decide his fate after the fall of Jericho). You can't tell me his high-tech scanners and such couldn't find a replacement object if he wanted to after Hank takes the quarter, be it a loose washer in the run-down church or a penny in the street or even something in the rusting-apart old ship full of scrap metal while he's doing recon. If shiny objects were interesting, there's bound to be something in one of those places he could pick up and fiddle with. Instead he only does it while he has downtime, and only before he deviates. The world is full of small shiny discs, and Connor can identify the chemical composition of drugs just by looking at them (even if that ability makes absolutely no sense, objectively). He'd find something if he wanted it, vs. it just being programmed activity.
