Premise: The adjectives intrinsic and extrinsic come from the Latin intrinsecus ("inner") and extrinsecus ("outer"), from the adverbs intra resp. extra, and the p.p. secutus of the verb sequor ("follow").
In the context of category theory, if we want to play the game of "non-philological (i.e. a posteriori) etymology", it is tempting to refer follow to arrows. I would therefore say that intrinsic is a categorical property of an object, which is stated by means of its only structure and self-maps, while extrinsic is a categorical property of an object which also depend depends from other objects and maps with different domain domains or co-domainco-domains. In this sense, compactness is an intrinsic property of topological spaces, while e.g. the homotopy extension property, or being an ANR, are extrinsic properties; in fact, most universal properties are. (But, I repeat, this is just a suggestion).