With custom workshops, we might be able to mod in a decent pottery system. Unfortunately we won't be able to scrape up clay like glassworkers scrape up sand, but if we make the workshop have tiles that need to be built on water (presuming we can do that) we could have a Dredge Clay reaction that makes lumps of clay out of nothing and rationalize it as having come from the water. (Alternatively or additionally we could have a Rehydrate Stone into Clay job that makes clay out of mudstone, siltstone, shale and the like.)
You could then have dwarves at the pottery workshop work the clay into a variety of drinking vessels and figurines. Probably you'd want to have them come out as, for example, a "greenware clay mug", which can then be turned into a "clay mug" in a Kiln or Magma Kiln. You could have another reaction that turns a "greenware clay mug" into a "glazed greenware clay mug", which can then be fired in the (Magma) Kiln to make a "glazed clay mug". (You could potentially have different glazes as well, its just a lot of copy and paste work.) The one unfortunate bit is that I don't think decorations applied early on will get carried over to the next stage, so you'll have to make sure you don't add menacing spikes to your greenware before glazing and firing.
Ideally the "greenware" "glazed greenware" materials would have a tag that would make them be damaged or destroyed by water, but I'm not sure if we'll have a tag for that. Unglazed clay is porous, so maybe we'd want to require that containers to be used to carry water or booze be glazed before an option to fire them is given. (In theory unglazed urns would make sense as no-liquids-allowed barrel replacements, but I think barrels are probably hard-coded to hold both food and drink at the moment, and I'm not even sure if anyone has figured out how to get barrel-like items from reactions.)