This is a more difficult issue than one might think. It could actually be wrong-headed to try to identify Confucianism in any definitive essentialist way. Why? Because the tendency, as with any “-ism” would probably be to try to nail down a certain number of doctrines or ideals that should be considered orthodox. That might be a legitimate way to identify some “-isms” or “-ities” (as in “Christianity”), but it should be regarded as a particular approach among others. So, let me suggest that trying to answer the related question: “Who is a Confucian?” is more fruitful. There are a few ways to go about doing this that differ in emphasis, though they might not produce mutually exclusive identifications:

1. Allegiance to Confucius – One way to identify Confucians is to see who glorifies, canonizes, or categorically defends and promotes teachings attributable to the historical figure known as “Master Kong” (孔子, Kongzi; 551 to 479 BCE). Here, the emphasis is on the unique identity of some historical person to whom “Confucius” refers. This approach should mostly end up including the ritual and spiritual devotees of Confucius–people who in various ways practice the devotional worship and study of Confucius (which still flourish in S. Korea and Taiwan). But it should also include anyone else, including traditional commentators and contemporary scholars, who is invested heavily both in the truth of and authenticity of certain teachings regarded as having issued from Confucius’s own mouth. Here, it actually matters that–or whether–Confucius was a sage, someone who not only taught things but embodied and exemplified something uniquely admirable. By the same reasoning, it matters whether the teachings traditionally attributed to Confucius are really his and not merely some good ideas that anyone could have come up with.
2. Advocating a Set of Uniquely Specifiable Ideals – This way of identifying Confucians focuses on espousers of the doctrines or concepts that can be specified as belonging to the unique set of views that arise from the moral, political, and (roughly) spiritual tradition that claims Confucius as an early founder. Here, the emphasis is on the strength and viability of a certain set of ideas, regardless of how historically accurate it is to attribute them to Confucius or any other particular individual in the tradition. A Confucian in this sense believes in the non-parochial defensibility of the value of a certain type of society based on the model of family affection and duties, certain types of bearing and decorum in public and private life based on self-reflection, and inculcation of certain feelings and traits through education. It is important to emphasize that a Confucian in this sense regards a core of such ideas as defensible, adaptable, and worth philosophical attention from reasonable points of view that are not already committed to the Confucian tradition.
3. Rearing in a Particular Type of Society – This way of identifying Confucians emphasizes participation, either witting or unwitting, in a type of society regarded as culturally bound in some non-trivial way to mores and institutions from a Confucian past. “The Confucian past” may be identified as the historical eras in which scholars and ministers with certain sorts of Confucian textual allegiances (see 1 above) influenced policies, laws, and ideologies that played a large role in shaping cultural attitudes. So, someone reared within a very traditionally Chinese or Korean subculture may carry such attitudes with them, whether they realize it or not–having a strong tendency to be scandalized by perceived indecorous behavior or appearance, for example.

Okay, let me leave it there for the time being. I’m sure there’s plenty more I would need to say to clarify things. Feel free to comment or question, name names, point fingers, etc.