Most of this discussion hinges on the precise definition of the word, “common.”
One definition, and the one most here are using, is, “a frequent occurrence, happening with regularity.”
Another definition, and, I believe, the one envisioned by the coiners of the phrase “Common Era,” is “a connection between two or more disparate systems.”
Many cities and communities have a “Commons,” or an area or park, usually near the center of the village, where people from different areas of town can get together for a variety of purposes. Many industrial systems have automated alarms, and several functionally-related conditions will set a “Common” alarm, which may or may not happen with any frequency (it may, actually, be quite rare.) Electrical systems occasionally have a “Common Bus,” where several different logical circuits share the same electrical path.
In the case of the “Common Era,” I, and many other people, perceive it as an adaptation to the multitude of local, incompatible date-reckoning systems which had been in place at the time.
It had been a long-established practice to date noteworthy events from the beginning of the reign of the local monarch, warlord, or ruler. This, unfortunately, meant that people from different regions would have widely varying dates for the same events. It was the intention to reconcile these incompatible systems that led to the introduction of a “Common” dating convention, whereby all “civilized” peoples (i.e., those who we communicate with on a regular basis) would be able to agree on a calendar.
Since the Roman Catholic Church had spread throughout Europe at the time, and virtually every community had a representative who spoke and understood Latin, it was obvious (to those who instituted the calendar) to use the purported date of the birth of Christ as the starting point of the new calendar.
As Buck Godot pointed out, we currently use the Roman pantheon to name some of our months, the Norse pantheon for the names of our days, and nobody feels we are promoting belief in these religions thereby—why cause a major international disruption in the naming convention for our years, to preserve the tender sensibilities of a few who find “Common Era” distasteful?