Direct answer first: No. We do not always see the same fraction of the Moon’s illuminated side. The portion of the Moon that appears lit from Earth changes over the month due to the Moon’s orbital geometry around Earth and the Sun’s illumination; however, because the Moon is tidally locked to Earth, we always see the same near side, while the exact amount of illuminated surface visible from Earth varies with lunar phase and libration. Explanation and details

• Synchronous rotation (tidal locking)
  • The Moon rotates on its axis in the same time it takes to orbit Earth. This means one hemisphere (the near side) always faces Earth, and the far side is hidden from view. The basic reason we can’t ever see the far side face directly is that the Moon’s rotation period matches its orbital period around Earth. This is why we consistently observe the same hemisphere from Earth [web results on tidal locking and synchronous rotation].
• What we actually see changes over the month (phases)
  • The Sun-Earth-Moon geometry changes as the Moon completes its roughly 29.5-day cycle, producing new moon, crescent, first quarter, gibbous, and full moon phases. At different phases, varying fractions of the Sun-illuminated portion of the Moon face Earth, so the visible illuminated fraction grows or shrinks accordingly. The maximum fraction of the Moon’s near side that can be illuminated at once is less than 100% (the full moon shows the entire near side lit), and at other phases you see less, depending on the angle between Sun, Moon, and Earth.
• Libration allows a bit more than half over time
  • The Moon’s slight oscillations in orientation relative to Earth (libration) let observers glimpse up to about 59% of the Moon’s surface over time, even though the near side remains the same hemisphere facing Earth most of the time. This wobble is due to the Moon’s orbital inclination and its axial tilt relative to its orbit around Earth. So, while the same hemisphere faces us, we can see a little more than half of the Moon’s total surface across all observations, but not the entire far side at once [libration concept].
• The “same amount” question in practice
  • At any given moment, the illuminated portion visible from Earth depends on the Moon’s phase. For example:
    • New Moon: almost 0% of the Moon’s near side is illuminated from Earth.
    • First Quarter: about 50% of the near side is illuminated.
    • Full Moon: nearly 100% of the near side is illuminated.
  • Across regions on Earth, the exact portion visible can vary slightly due to parallax and the observer’s location, but the governing factors are phase and libration, not a fixed equal illumination across all times.

Bottom line

• We never see the far side from Earth due to tidal locking, but we also do not always see exactly the same illuminated fraction of the near side because the Sun-Moon-Earth geometry changes through the lunar cycle, and libration lets us glimpse slightly different regions of the near side over time. The combination of these effects explains why the visible lit fraction varies across the lunar month, even though the Moon presents the same face most of the time.