The establishment of the identity of particular properties (SEP) is a matter of metaphysical inquiry. For instance, properties themselves have properties, and those properties are used to determine if two properties are the same property, or differ in some way. Terms like co-intensional, co-extensional, intrinsic, extrinsic, and natural properties (SEP) are some terms to describe different types of properties.
But, setting aside, a deep philosophical analysis, how do we generally know if two properties are of the same sort? This can be achieved in several ways.
First, intuitively. It is not unreasonable for a person to say this house looks green, and this tree looks green, so they share the property of green. Children do it everyday. Thus the property of greenness is sufficiently established this way.
At the other end of the spectrum, we have properties that are not immediately empirical and are highly abstract, and exist because of operational definition. Thus, for a psychologist to establish two people are both suffering from vulnerable clinical narcissism means defining it in some specific way, and then administering a psychometric, for instance, to be assured that two people qualify as exhibiting the property of having a specific personality disorder. Those criteria are laid on in the DSM V, and the psychometrics administered help support that conclusion.
Properties may be labels for dispositions or for occurrences. Dispositions are tougher to establish, because they imply a regularity or pattern. For instance, just because someone has killed another human being is not sufficient for entailing the property of being a psychopathic, serial killer. Thus, there may be some degree of justification involved in make a determination of a property, such as meeting a definition set about by criminal law.
Science attempts to use empirical evidence to establish equivalence. For instance, to determine if a house and a tree are both actually, green, optical equipment may be used to measure wavelengths and satisfy an optical, abstract definition of green that goes beyond mere first-person observation. The advantage of this sort of establishment of establishing if two properties are the same is that it avoids difficulties in observer bias. One of two observers, for instance, may have red-green color blindness, in which case, it may be the case that the house is actually red, and despite that it appears to such a person both the house and the tree have the property of greenness, it isn't the case objectively.
Natural kinds (SEP) is a broader discussion than just the ontological, because it not only relies on discussion about properties, but more broadly epistemological, and other metaphysical conceptions (such as metaphysical necessitation) to make a claim about the natural structure of the world, and is therefore a realist claim (as opposed to instrumentalism). Therefore natural kinds revolves not so much around questions of properties and identities (such as Leibniz's Identity of Indiscernibiles), but more about understanding the broader question of how language, knowledge, and things in nature interrelate. From SEP:
This article divides philosophical discussions of natural kinds into four areas: metaphysics, epistemology, philosophy of science, and philosophy of language. The metaphysics of natural kinds asks whether we should think of our supposed natural kinds as genuinely natural. And if they are, what are natural kinds? And, finally, do natural kinds have essences? Philosophy of science is concerned with natural kinds because, as mentioned above, it is the use of natural kinds by the individual (‘special’) sciences that generates our interest in them. So we may ask, whether the kinds appearing in our best scientific theories do indeed satisfy the theories of natural kinds proposed by the metaphysicians.
Thus rejecting natural kinds may focus on showing that the construction of language and the use of linguistic categories is somehow arbitrary, and that a chicken doesn't necessarily have strict boundary conditions, but rather chickens are birds and birds are almost reptiles, and as the genes drift, there may not be strict categories in terms of science, but rather are conveniences of cognition. For instance, Stephen J. Gould once declared there was no such thing as a fish (PDF). Understanding biological categories as Aristotle understood them, and as modern evolutionary biologists who use clades do is an entirely distinct affair.