It is raining is how we normally describe the weather on a rainy day. However, that doesn't make rain is falling grammatically incorrect. That construction may not be a common way to describe the weather, but it's not "wrong."
The word rain can be a verb, or a noun; as a noun, it refers collectively to raindrops. Moreover, falling can refer to anything dropping from the sky. Therefore, paratroopers can be falling, drones can be falling, and rain can be falling.
One might regard "rain is falling" to be a more poetic way to talk about the rain; for example, "Rain is falling in Santa Fe" might be considered more poetic than "It is raining in Santa Fe."
Here's another place where I could see a writer using this way of phrasing it:
The firefighters had been battling the wildfires for three days. On Friday morning, their prayers were finally answered; rain was falling as the sun came over the mountains.
Bottom line: Don't confuse "less common" with "incorrect".