What is the difference between an object K and the thing which satisfies ( is an extension of', 'is an instance of' ) the property P ( or 'universal', 'predicate' , 'qualia', 'concept' ) :
P(x): x is K
One may formalize this is any formal system with a good concept of equality. One of these is homotopy type theory which knows the identity type (X = K). It is then a theorem that Sum_X (X = K) is a contractible type whose essentially unique inhabitant is K itself.
David Corfield just the other day wrote a note on this for an audience of philosophers interested in structuralism.
For more along these lines see also the references at nLab:structuralism.
J. Hintikka has written a classic paper on the problem, which he calls the Fregaean trichotomy: 'is' as predication, identity, existence.
Hintikka J., "Is", Semantical Games, and Semantical Relativity, J. Philosophical Logic, Vol. 8, No. 1 (Jan., 1979), pp. 433-468 (Springer|Jstor)
After discussing these aspects and adding more, he concludes: the trichotomy is probably wrong and notes that "Frege, Russell, Quine, Davidson, Chomsky, Lakoff all were mistaken".
The topic defies surveying but the Ontology site has compiled a bibliography https://www.ontology.co/pdf/existence-predication-biblio.pdf.