It’s not a god per se, rather the animal companion of Ganesh (Elephant god/ god of people, kinda like god Fufluns of Populonia). The mouse is like a ride of the god.

And maybe the physical thing you provide to that person may only last for an hour, but the memory of something a kind person did for them will (hopefully) persist as well

I don’t know if you have been in India but I have. Begging has become a profession there. So most likely they won’t recognize individual donors. Also what OP mentioned is a real problem. Easily you’ll meet more than 20 homeless people at once around you in any big railway station.

From what I understand, shitty politics and rising inequality is the reason behind this.