Elimination by contrast:
Instead of tracing the Latin root first, test the four options against what "sharing together" implies. Apathy means indifference, the opposite of engaging in a shared exchange, so it cannot describe the goal of communication. Differences, by definition, is what communication tries to bridge or reduce, not what it is "for" - so it does not fit either. Popularity is a possible side effect of being a good communicator but is not the reason communication exists.
What remains is commonness: when two or more people share information, they move toward common understanding, common meaning and common ground, which is exactly what "sharing together" describes - confirming option (1).