Flour – Plain flour or strong bread is there any difference when making a cake

"Strong flour" and "Bread flour" generally mean the same thing -- lots of gluten, so the dough can stretch and incorporate lots of bubbles.

Not all bread demands high-gluten flour, but the traditional airy loaf of Western Europe and most of the USA does.

If what you're making is bread of that kind, "strong bread flour" is what you want.

If you're making a cake or a sauce, things will probably be fine -- try it. It might not be perfect but it won't be a disaster. It's worth having some less strong flour (often labelled as "plain flour") in your store cupboard.

The author of this recipe probably happens to keep in his pantry (or more professionally speaking, dry store) just those two types of flour, and so has specified a mix to get a mid-level flour with moderate protein levels, tailored to his preferences. Given that the author is Jacques Torres, this is almost certainly a scaled down translation of a professional recipe, where that is not an uncommon practice (the very odd measurements support the idea of scaling and rounding; the weird flour measurements are probably the closest volume equivalent to a weight based scaled recipe).

Two commercial varieties (all purpose, and the less common pastry flour) also have protein levels in between cake flour and bread flour, with pastry flour being somewhere between cake flour and all purpose, typically.

If you happen to have all purpose (and you almost certainly do) I would suggest using it in the recipe for the total of both specialty flours. Very few cookie recipes are so fussy that it will actually matter very much.