Baking it won’t eliminate the oils or old spices, more would give you your cast iron frying pan effect.
We use a super neutral dish detergent that washes or at least soaks out in rinse water. Not one of the national name brands.
Even were this cast iron, sometimes you get to the point that you have to clean and restart to build the finish.
But others may feel differently.
I was thinking through what would happen should the OP follow the advice by another user which recommended baking the mortar and pestle.
Since it has a heavy film of fats,my thought is that baking at a low temp would create a finish similar to that on seasoned cast iron. I’m not thinking that would be a plus but others might think otherwise.