The stag is an Easter egg that can be found in three specific locations in the Forest.
Though the locations where the stag will appear are set, the chance of the stag actual appearing is random. Each time Young scrolls a screen into view where the stag can appear, there is a 1% chance that the stag will appear. Alternatively, you can hold Q, W, and E to force the stag to appear.
Young cannot interact with the stag, and the stag will remain in place until Young exits the screen it appears on.