No, not all moons orbit at the same rate as their planet. The orbital periods of moons around planets can vary depending on several factors, including their distance from the planet and their initial conditions at the time of formation.
In general, the orbital period of a moon is determined by its distance from the planet and the planet's mass. Kepler's third law of planetary motion states that the square of the orbital period of a moon is proportional to the cube of its average distance from the planet. This means that moons farther from their planet generally have longer orbital periods.
However, other factors can also influence the orbital period of a moon. For example, gravitational interactions with other moons or nearby celestial bodies can perturb a moon's orbit and cause variations in its orbital period. Additionally, tidal forces from the planet can also affect the orbital period of a moon over time.
As a result, while some moons may have orbital periods that are synchronized with their planet's rotation, such as the Moon with respect to Earth, many moons have different orbital periods and may exhibit various types of orbital resonances or irregularities.