Could depend on what's in the local tap water (mineral content as well as pH), in addition to lighting as weeeeum mentioned. I have some scallions on my planted aquarium (no fish, but I lightly fertilize for the plants) with a grow light, and I can harvest one of the four plants once a week. So even with active grow light and fertilizer, the once a month per plant statement seems right. Also note I started with seven plants, all from the same store-bought bunch, but three of them died over the first couple of months.

I don't think they get enough sun. Other than that they were in a cup at the kitchen window, with the water replaced every couple days.