It looks to me like calcium or mineral deposits from water leaching thru the brick. Could there be a pinhole water leak in the roof or down spout?

Definitely mineral-based: the roof over the exterior wall, maybe a rusted screw on your gutter, OR possibly the previous owner had a plant there. You'll need lime-away toilet bowl cleaner and depending on how much damage has been done to the grout, I use either a nylon pot scrubbing brush or a copper-bristled parts brush.

How long have you owned the house?

That downspout looks way younger than the mineral spot. If it's not growing, then the previous owner fixed the moisture issue with that *ugly as sin* drain extension. Personally, I'd have replaced/sealed off the existing downspout- then put 3 T's in to the gutter equidistant along the walkway, Japanese gutter bells, and a French drain in.

https://rainchainsjp.com/products/migumo/

https://rainchainsjp.com/products/migumo-l/

Is the fountain functioning? It looks like there are some flat pavers up close to the house; I wonder if the fountain was sitting closer to the house and splashing against it at one time. The shape of the stain mimics the shape of the fountain. If the birds bathe in it, they can splash a great deal of water outside of the fountain. Or perhaps there was a pot there which contained a plant that caused the mold on the wall?