따지다 is picking something apart and reckoning it strictly. Seven of these eight lines are the same phrase — 그렇게 따지면, "if you're going to reason like that" — used to point out that someone's logic proves more than they meant: 그렇게 따지면 다 같은 소속이네요 "by that logic, we all work for the same company". It is a mild challenge rather than an insult. 세세하게 따지다 is going through something in fine detail.