In Japanese, a gloomy/depressed person can be referred to as being "damp" (じめじめ, jimejime). What grows in dark (i.e. gloomy) and damp places? Mushrooms, of course.
The expression from Haruhi there about telling Tamaki to "stop growing mushrooms in one's closet" which means, more or less "stop being depressed".
They're called Al-mi'raj. And they have been around for hundreds of years.
Dragon Quest is only the first game to include them in.
Series could just be using them as a starting monster as a simple DQ reference... But it could also be because they're like one the most easiest fantasy animal to kill. The most they can do is scare off wild animals.