My enthusiasm was somewhat dampened, however, by a soon-to-be-published study by J. Robles and colleagues where they performed a comprehensive comparison of the Sun to other stars. In this study the authors established 11 relatively independent star properties that are thought to affect whether a planet orbiting the star can support simple (as opposed to advanced) life for a short time (as opposed to simple life for billions of years). In such cases, the likeness of a supposed twin can be more relaxed.
Ideally, one would like to measure these properties for a large collection of stars in order to obtain the distributions in each property. The Sun would then appear as a point in this 11-dimensional distribution and measurements would show whether the Sun was out on the edge of this distribution (indicating rarity) or somewhere in its middle (indicating ordinariness). Of course, to get a proper measure of the uniqueness of the Sun, some kind of a chi-squared statistical test should be applied, from which various probabilities can be derived.
Unfortunately, the number of stars possessing all 11 of the properties is very limited, so the authors of the study had to use limited sets of stars that were parts of other surveys, and in many cases these sets were not overlapping. To see what this limitation could do to bias the study, imagine a comparison for just two properties with the stars being completely nonoverlapping so we don’t know if the stars in one set have skewed values for the other property. It is possible for the Sun to appear in the middle of both property distributions but not be in the middle of the joint distribution for stars where we can obtain both properties. In practice the nonoverlapping distributions are not likely to be so skewed, but it does reveal a weakness in the conclusions of the study.
The authors conclude that the Sun is not unique. If the study is free from the non-overlapping bias, one would expect to be able to find stars that are twins of the Sun in many, if not all, of the 11 properties. However, based on the studies Hugh Ross mentioned above, finding such a twin, even for a limited number of properties, has proved difficult, at least for the more restrictive advanced-life case.
Perhaps in the end the Sun will not prove to be unique, but based on the results so far, that appears to be the case. Future deep surveys with accurate measurements for all the relevant stellar parameters hold promise to help settle the matter.