No, it's not bile. It was a poorly preserved pig, and the brown, icky stuff is old blood that didn't flush out with the fixative. Run it under gently running cold water to wash it all out, set it on its "belly" for a few minutes to let the rest of the water drain out of the incision, and you'll be all set. I can't believe your teacher just dismissed the question with "sure, why not" without even looking! This is why people end up not learning biology right, because some teachers just can't be bothered to give a proper answer. Oh, and the blood probably didn't originate from cutting through the liver, even if you hadn't hit the liver, you'd have found it filling the abdominal cavity...when they inject the latex into the veins and arteries (so they'll be pretty pink and blue for you), some of the vessels rupture and you get either the blood in the cavity, or if you're really unlucky, you get a pig with a belly full of latex...those are horrible to dissect and you can be pretty sure you're not going to be able to identify all the blood vessels or rely on them being the color they are supposed to be. Also fun when someone switches the colors of the latex going into arteries and veins and all the color-coding is backward of what your lab manual tells you it should be.
Um, yeah, I've taught a "few" pig dissection labs...there are all sorts of fun oddities you come across aside from the usual anatomical variation you're supposed to see. If you think it's hard identifying organs in the pig you're dissecting, it's much harder identifying organs in a pig someone else is dissecting...especially when the ones that are supposed to be in the pig are already somewhere out on the dissection pan. It's all fun though. Just eat a good meal before class and make sure they keep the windows open or have you working in fume hoods so the paraformaldehyde odors don't overwhelm you (for people prone to low blood sugar, something about those fumes can trigger fainting if you don't eat before class...it's really not just squeamish people...you shouldn't have to breathe those fumes anyway).