When it was first introduced in the Celestial Update, changing characters using a Moon Rock Idol reset the player's Map, Day count and Crafting recipes. This is no longer the case as of update 288118.
The Idol glows when it is in proximity to the Celestial Portal.
An Idol is defined as an image or object that is used to address deities or spirits with devotion.
It gives off a small amount of light at night, but is not an appropriate light source to ward off Charlie.
If Woodie offers a Moon Rock Idol to the Portal, Lucy the Axe will turn into a regular Axe when returning as a different character.
When the Moon Rock Idol is offered to the Celestial Portal, the Character will perform an animation similar to the one used for The Lazy Deserter.
The unique Character quotes when offering a Moon Rock Idol seem to indicate that using the Celestial Portal is similar to the classic description of death, in which one is said to see a bright light.