The amount of oxygen in the air is the same day or night. The big difference - which isn't really explained in the part of the PHAK you quoted - is that at night your eyes work differently.
Your eyes have two different types of photoreceptors: rods and cones. Cones work best with high light levels and rods work best at low light levels. So during the day, when there's plenty of light, you're seeing primarily with the cones. At night, it's the opposite: your vision depends mostly on the rods.
But, rods are very sensitive to oxygen levels. As the level decreases, they quickly stop working effectively. During the day this doesn't matter because you're not using them anyway, but at night it does matter and you can lose night vision at relatively low altitudes. This is why the PHAK recommends supplemental oxygen at night: it's to make sure that the rods continue to work effectively.
The PHAK has a lot of information about "vision in flight", starting on page 17-19. Page 17-24 explains the need for oxygen at night:
Unaided night vision depends on optimum function and sensitivity of
the rods of the retina. Lack of oxygen to the rods (hypoxia)
significantly reduces their sensitivity. Sharp clear vision (with the
best being equal to 20–20 vision) requires significant oxygen
especially at night. Without supplemental oxygen, an individual’s
night vision declines measurably at pressure altitudes above 4,000
feet. As altitude increases, the available oxygen decreases, degrading
night vision. Compounding the problem is fatigue, which minimizes
physiological well being. Adding fatigue to high altitude exposure is
a recipe for disaster. In fact, if flying at night at an altitude of
12,000 feet, the pilot may actually see elements of his or her normal
vision missing or not in focus