Answer

It’s hard to give a clear answer here. Simply speaking, holiness is a law that applies to the object itself, and that means there is something in the thing itself. Our attitude toward it follows from its holiness, but that attitude is not the essence of holiness. Although many have argued that the second possibility is the correct one. This is connected to Yohanan Silman’s articles on nominalism and realism in Jewish law, and in particular there is an article by Henshke on Maimonides’ approach to the question of impurity and holiness. As I recall, he argues that these are laws that apply to the person rather than to the object, but I think he is mistaken (I also remember not being convinced by his proofs).
Leibowitz always claimed that this is a law applying to the person, and his proof was from Mishnah Kelim chapter 1, for example regarding the holiness of the Land of Israel: “And what is its holiness? That the omer and the two loaves are brought from it.” But this can easily be rejected, since the Mishnah there merely gives halakhic implications (for all ten levels of holiness) and does not necessarily define holiness itself that way.