Walls don't need to breathe--people do! What these people mean is that it is important for walls to be able to dry out when they get wet; that they should not trap moisture. This is true, but is only orthogonally related to "breathability." In practical terms, what this means is that in reasonably moist climates, you want a wood-framed house's wooden sheathing to be able to dry to either the interior or the exterior if it gets wet. The sheathing would dry to the exterior if it's placed closer to the siding than the interior drywall, and to facilitate this, you would want there to be a ventilated rainscreen gap between the siding and the tar paper or housewrap covering the sheathing. If, on the other hand, the sheathing is covered up with a bunch of rigid foam insulation, it needs to be able to dry to the interior, which is accomplished by not putting any totally impermeable materials between the drywall and the sheathing like plastic sheeting or closed-cell spray foam.
As to how this relates to the siding material: it's irrelevant. Vinyl siding in no way impedes a wall's drying power, and in fact often improves it because it automatically includes a weak form of rainscreen gap in the form of the open channels behind the planks.
On a wood-framed home, you should choose brick as an exterior cladding because you like its look, durability, low maintenance, or acoustic advantages, not because of its moisture-related characteristics. In fact, brick is more dangerous in a cold and rainy climate because the brick will soak up water, which can freeze if the temperature drops far enough before the water has all drained out of the brick. And the brick siding absolutely needs a ventilated rainscreen gap separating it from the tar paper or housewrap covering the sheathing. Otherwise a phenomenon called "inward solar vapor drive" can force any water in the brick into whatever's touching behind it--such as the wooden sheathing!