The Christian Register is older than the organization most people would call its publisher. David Reed put out the first issue on April 20, 1821, four full years before Boston area liberal ministers ever sat down to organize the American Unitarian Association. The paper was already running when the AUA was born, not the other way around.
What happened next is the part that matters. Within a year of its own founding, the new association's executive committee took the paper over. Its own 1826 annual report says so directly: the committee had "undertook the conduct of the Register at the commencement of the present year." From that point the Christian Register was not merely sympathetic to Unitarianism from the outside. It was the AUA's own weekly, running the words of William Ellery Channing, Henry Ware Jr. and Edward Everett among others, for the next 131 years.
The paper only stopped being the Christian Register in 1957, renamed The Unitarian Register as the denomination grew, in its own words, "less and less focused on Christianity." Four years later it merged with the Universalist Church of America's own newspaper, the Universalist Leader, in exactly the year the two denominations themselves merged into the Unitarian Universalist Association. A publication founded before its future publisher even existed ended up outliving that publisher's separate identity by exactly as long as it took the two churches to become one.