Why do some people think that the moon is a star or a planet?

Greek: "aster" means any object in the "heavens" (on the celestial sphere)
From the word "aster", we have the English word "star".

The moon is in the "heavens". For Greek astronomers, it was definitely a star.

"Aster planetes" (a star that moves, in relation to fixed stars, in a predictable way).
In English, we get the word "planet" from the Greek phrase "aster planetes".
For Greek astronomers, the Moon was a "planet" and remained a planet, even in European astronomy, until the late 1600s (around 1687)

The name of our satellite is "Luna" (with a capital M). The simile "moon" (small m) can be used for objects that are "similar to the moon".
Americans still write "moon" (small print) because some American astronomy magazine made the mistake of insisting that it should be written that way (a few decades ago).

When Galileo discovered the "moons of Jupiter," he called them planets because they fit the definition as it existed back then (they saw each other in the heavens and moved predictably between the fixed stars).
The planets included the Moon and the Sun (also an object in the heavens, an "aster", which moved between the fixed stars)

Only after Newton offered proof that the Sun was the center of the planetary system (1687), new words were introduced.
The Sun and the Moon lost their status as planets.
The Moon (and things around Jupiter) became "satellites".

Astrology (which was already condemned by Greek astronomers as false, for 2000 years), continues to call the Moon and the Sun "planets," because astrology in the Western world began long before 1687.

In astronomy, the Moon is a satellite and its name is written with a capital M (at least outside the US).
In astrology, the Moon is a planet that turns out to be a satellite of Earth.

For most people, the Moon is simply what they see at night, shining brightly.