I've seen these clay bowls with lines before and I knew they were some kind of mortar. In Hidalgo and Oaxaca, they seem to be called molcajetes, just like their volcanic rock cousins. I've also seen them called chilmoleras, which makes more sense (chiles and molinar).
In Muñoz' Diccionario Enciclopedico de Gastronomia Mexicana, he suggests it's especially useful for the classic Yucatan dish chilmole.
The issue is what to use for the pestle! I have a wooden pestle from a Spanish mortar and pestle set that I'll try out.