Massachusetts does not lend itself well to the typical suffixes (-ian, -er, etc.) attached to the name of a place to refer to someone who is from that place (perhaps because the name is not of English origin.) Thus, commonwealther is used as a reference to the fact that Massachusetts is officially designated as a Commonwealth instead of a State (although there is no legal distinction between a “state” and a “commonwealth” in the United States.).
There are three other "states" in the US that designate themselves as Commonwealths (Kentucky, Pennsylvania and Virginia.) However, the usual suffixes are applied to them (Kentuckian, Pennsylvanian and Virginian) without issue, and so a new word is not needed to describe a resident of one of those states.