:)
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 >
I’m not real big on Language based on usage. There are rules and laws for language just as with life, etc. One reason, I might add, that the Constitution has managed to assume the toilet tissue role it has managed over the years, not based on law, but on usage or lack of.
At least when I grew up the rule was, the noun/subject got the plural.