Whether Confucianism is a religion has been argued for four hundred years, and the argument has never been merely terminological.
The case against is that Confucius said little about gods and less about what follows death, declining questions on both; that the tradition's concerns are ethical, educational and political; that it has no creation account, no salvation and, for most of its history, no separate priesthood; and that its Chinese names (rujia, the school of the ru, rujiao, the teaching of the ru, ruxue, the learning of the ru), do not map onto the Western category of religion at all, that category being itself a comparatively recent import into Chinese usage.
The case for is that Tian, Heaven, functions as a moral order that grants and withdraws sanction; that the Mandate of Heaven made political legitimacy a matter of Heaven's approval and its loss a matter of Heaven's withdrawal; that the ancestral rites are performed as rites and are addressed to the dead; that Confucius himself was honoured in temples with state sacrifice for centuries; and that the reasons usually given for excluding the tradition often amount to observing that it is not much like Christianity.
The question has had practical consequences. Jesuit missionaries in China from the late sixteenth century argued that the ancestral rites and the honours paid to Confucius were civil observances that Chinese converts could continue. Rome eventually ruled against them, and that prohibition stood until it was withdrawn in the twentieth century; the episode is known as the Rites Controversy and it cost the mission heavily.
The label applied to the tradition here, Chinese ethical-religious, is a compromise and should be read as one. It records that the tradition does not fit either category cleanly rather than settling which one it belongs to.