In a camera, light bounces from a light source, off an object, into receptor, with at least one aperture in the path to ensure that light arriving from different directions reaches different sensors. These sensors collect light from some area over time which can be recovered as information (chemically or electrically). Light passing through an aperture is subject to diffraction (which rules out a perfect square beam of light) and possibly other distortions, and the sensor may or may not be a perfect, uniform square, but in any case it's all equivalent to a perfect point in space that receives undistorted, refractionless light then applies a filter and integrates it over time and area.

The underlying physical phenomenon that allows integration to happen is the ability for a film grain or ccd cell to count up incoming photons.

These ideal sensors may be arranged on a grid, but the filter that maps source light to sensor value won't look anything like non-overlapping square boxes.

We are discussing the mathematical model of the system that has been found to be the most proper to deal with the practical problems that arise about image processing.

When he says "the device integrates over..." what he means is that we can model the display device as a reconstruction filter integrating over the point sample lattice.

Remember, we are discussing the mathematical model of image processing.

In the purely physical model the pixels in the memory are probably a linear array of n-bit numbers, that drive voltages over the illuminants of the display. However, the "linear array of n-bit numbers in a framebuffer" is a really awkward model to start discussing e.g. image processing algorithms.

The point sample model, as the author tries to persuade, provides a simple formalism for a lot of practical problems and is also, from the point of view of sampling theory, the correct model.

This paper only mentions integration in the context of sensors, not displays. So it's pretty straightforward: the scanner/camera/eyeball is integrating photons over time and over the surface of the sensor. None of those are inherently discretized by little squares, because even the photoreceptors on a chip aren't perfect rectangles and are subject to optical effects like things being imperfectly focused.