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When they cook some type of fat/oil such as lard or coconut oil plus a liquid with basic ph such as lye water, either from KOH or NaOH together, then they get soap. These are the two ingredients of soap. Nothing else is necessary.
Yet, when you buy a bar of soap in any shop/supermarket/etc and you read the ingredients you get a long list of gibberish. Usually they add stuff like perfume. But all these additives make impossible to use soap for cleaning the teeth. They tastes horrible and irritate the whole mouth cavity.
This is why I called the first thing a "pure" soap. Because it consists purely from the main ingredients.
It is acceptable if it contains salt as well. Salt is used to harden the soap. When a soap is cooked fresh, it has the consistency of a pudding, perhaps a bit more liquid. They leave it to dry for about a month before it can be called a "bar". But depending on factors this could still be too soft, so during the cooking process, they add salt to it, which helps to make the end product more dry.
As I mentioned in an average supermarket or a toiletry shops/drug stores you won't find "pure" soap, just commercial ones. They don't sell pure soap produced on industrial level anymore, they only use that as ingredient in other products.
But there are handicraft shops where they do, because selling home made stuff is a market in itself these days. Best bet is online, all the shops have online stores now. Have to make sure the ingredients are just that (perhaps they add essential oils for the scent but it might be still fine with that).
If you have more questions, ask.