The standard role of intuitive numbers in the foundations of mathematics

The usual answer (or dodge, depending on yuor philosophical position) is to talk about the intended model (more precisely, intended interpretation) of the natural numbers. If such a thing exists (and many mathematicians do believe it exists, including the constructivist Errett Bishop), then one can interpret references to "finite" at the meta-level as referring to things equinumerable with individuals in the intended model. If you don't want to believe in an intended model then you also have to give up hope of an absolutely rigorous development of mathematics "from scratch". Whether or not you actually lose anything in the process again depends on your philosophical position. Kronecker, by the way, never expressed any opinion on the matter in writing. Whatever is reported in his name is hearsay based on Weber (who certainly made a mistake when he mentioned "whole numbers" rather than "natural numbers").

Juat out of curiosity, I looked up "intended interpretation" on wiki, and was led to the following comment:

Intended interpretations. Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended to be the set of natural numbers. The intended interpretation is called the standard model (a term introduced by Abraham Robinson in 1960).[8]

Note the definite article, which I guess begs the question (namely, yours). Here the reference is to the paper

Roland Müller (2009). "The Notion of a Model". In Anthonie Meijers. Philosophy of technology and engineering sciences. Handbook of the Philosophy of Science 9. Elsevier. ISBN 978-0-444-51667-1

which you may find useful (though I hasten to admit that I never read it). If you gain some insights do let me know.

The book that the article is in can be found here.

For a related question see What are natural numbers? where you will also find an accepted answer containing a passionate defense of the intended interpretation without mentioning the term ("categorical" and all).

For a detailed discussion of the intended interpretation see this post.

George Reeb's position was that the naive counting numbers do not exhaust $\mathbb N$; see this post for a bit of a discussion and this article for more details.