The noun (mother) is what the plurality is, therefore it’s motherS in law. Think attorneys general, surgeons general, and the like. The adjective phrase (in law) remains the same, because there are not more laws being generated by the union.
Your referenced site is, to be blunt, wrong.
Just because “everyone” does something does not make it correct.
Don’t get me going about personal pronoun usage! < LOL >
moth·er–in–law - noun
plural: moth·ers–in–law
Some people might make a distinction between how they talk and how they write. When writing people should employ correct grammar, usage, etc.