Rain is an important atmospheric addition to conveying a particular set of emotions within both the world and narrative; when you imagine the clouds coating the sky in a rich grey, with the rain falling heavily and obstructing your peripheral vision, you're given a sense of loneliness. A feeling of being lost, with the world unravelling itself a gentle way as you take each step further into it; the rain is a mask.
The film Blade Runner perfectly conveys this atmosphere of depression and loneliness through rain: Los Angeles is hidden away in a blanket of negative emotions; it's forgotten; lost; void of all hope. As for videogames, Silent Hill uses rain to distress the player, telling them that the environment is not safe in the rain; bad things are coming: you are not prepared to take on the horrors that hide within it.
Those two examples use rain to project the narrative onto the viewer on a more darker, emotional scale. In a way that we can connect our personal experiences with heavy rain to feel and sense the hopelessness the characters themselves feel. One might make the claim that this very way of storytelling is toying with our most basic emotions; and it is in a way that affects our senses on a personal scale: to us, rain is something we would try to avoid given our most primitive instincts would tell us that predators could be lurking in the rain, taking advantage of the fact that we cannot distinguish particular important features of objects and people that we otherwise could in the day. There's additional mystery added.
I might describe a wooded area as being 'rather dull, with very little signs of life within it'. But once I describe the wooded area as being 'a place of darkness, coated in a constant downpour of rain, with very little signs of life', the person reading would begin to let their mind wonder as it tries to guess what could be lurking within; what could be in that area, subjecting itself to such a place? The former description just projects a more boring emotion, making the reader imagine a simple, relatively dead woods.
I'll leave you with an opening paragraph from a short story I wrote, that had a strong focus on rain and loneliness:
"Ethereal raindrops cause speckles of iridescence on the large, all-seeing window to the city; the sprinkles of neon tones originating from an ever-glowing capital give a celestial ambience as they glitter luminescence into the dark, surreptitious room of the four hundred and fifty-third floor."