To the extent that the contents of this page are precise at all, they are speculative. As far as I know, nobody has formulated, much less proved, the logical consistency and independence results that would formally justify its claims.
If you are content with set-theoretic foundations for category theory, you may ignore the adjective ‘strict’ in ‘strict category’; in ‘strict functor’ it basically means ‘non-ana’ (the usual default, but not necessarily the best concept). If you are content with set-theoretic foundations using the full strength of the global axiom of choice, then you can even ignore the ‘non-ana’ bit.
A strict category is a category with a notion of equality (not merely isomorphism) of its objects.
In most foundations of mathematics (not only ZFC but also structural set theories such as ETCS and SEAR), every category is necessarily a strict category, because it has a set (or class) of objects, and elements of this can be compared for equality. However, some foundations (including FOLDS, SEAR+ε, the type theories of Martin-Löf and Thierry Coquand, and preset theories) allow one to naturally define the term ‘category’ so that a category is not inherently strict.
A strict functor is a functor between strict categories that preserves equality: if . Again, in most foundations, this happens automatically, and one must pass to anafunctors to recover a properly weak notion of functor (although the axiom of choice shows that every anafunctor can is equivalent to a strict functor). Note that a natural transformation between strict functors is automatically strict.
A 2-category must have strict categories as its hom-categories to write down the definition of strict 2-category; a similar remark holds for strict 2-functors. As such, strictness in the sense of this page matches and extends strictness in the usual sense of higher category theory. However, we now get a notion of strict -natural transformation as one that preserves equality. (All modifications between strict transformations are automatically strict … until we get to -categories, of course!)
Even if one's foundations allow one to notice the difference between strict categories and properly weak categories, the practical content disappears in the presence of the axiom of choice. (This is just like the equivalence between functors and anafunctors in that case.)