They are both grammatically valid, but you have to be aware of the logical context. They are both correct from a strict grammatical perspective, but the first one makes more sense and sounds much more natural in this particular construction. In the context of a comparison ("is better than") the reader needs to know what is being compared. When you use the plural, it sounds like you're comparing one group to another, and it is not clear what the group is.
Typically there is only one man and one woman who would be in a relationship at any given point in time, so the singular sounds better. When you say "no men are better than women at maintaining relationships", it implies there are multiple men involved in the type of relationship in question. Unless the context specifically relates to same-sex relationships or group relationships only, it's confusing and doesn't seem to fit the logic of the proposition.
There might be instances where comparing groups like that might make sense. For example, you could say "no Americans are better than Canadians at fielding a hockey team". The literal implication is that you could select any group of Americans and any group of Canadians and that would always be true, so probably it's logically false, but because people play hockey in groups it makes a little bit more sense. It's the sort of thing a drunk Canadian might say instead of "Canadians are usually better at hockey than Americans."
Where "no men" makes more sense. Outside of the comparison context, it is easier to find circumstances where "no men" makes sense and would be the more natural usage. For example, "no men are allowed into the women's restroom" sounds good and it is clear that you're referring to all men everywhere.
In general the construction "no man" sounds a bit like a proclamation. It is not unheard of for ordinary use, but perhaps more something a king or philosopher would say. So the phrasing "no man is allowed into the women's restroom" is fine in a technical sense, but sounds a little bit like a sign on a royal bathroom (or an English language learner). The phrasings "no man has set foot on Mars" and "no men have set foot on Mars" are both equally valid and natural. Because going to Mars is a grander undertaking, the more dramatic phrasing of "no man has set foot on Mars" becomes more appropriate than when you are discussing a bathroom.
According to Fowler's Dictionary of Modern English Usage, which is a standard grammar reference book, the compounds that derive from the word "any" such as "anyone", "any way", "anyhow", "anymore" or the rare, "anywise" are all singular.
According to the Oxford Learner's Dictionary, here:
"any" can be used with uncountable or plural nouns in questions
The Oxford Learner's Dictionary does recognize that "any" can be used with singular nouns when it does not matter which one, such as in:
Pick any card.
The website referred to in the query is correct.
There is a difference of meaning in the two examples stated in the query.
The first sentence,
If you have any questions, please let me know.
is something a teacher would ask a group of students in order to make sure an explanation is clear.
The second sentence,
If you have any question, please let me know.
would mean the same as the previous example if "a" was substituted for "any". But as the sentence stands, it is a little unusual. It could be understood to be regarding the questioning of a person's guilt or innocence. "Doubt" would be a better word instead of "question" in this context.
The word "question" in the second sentence could be understood to mean "controversy", but this would also not be the usual construction.
Best Answer
"Any" can be used with either a singular or a plural noun.
It's the same in an "if" clause. It depends on who you are talking to, and what you want to say. For example, all of the following are valid:
Note the difference in phrasing. If I want a single idea, I specify the idea (an idea how to ...). If I want multiple ideas, I instead have to specify the general topic (some ideas on how to ...
In your examples, you're asking the same question from a slightly different perspective:
Both are fine, however there may be a slight difference in nuance as the singular might emphasize these individuals have some unique quality. For example, imagine a drill instructor talking to a group of new soldiers:
Depending on the intonation, this can be seen as a challenge to some especially brave recruit. The plural might instead be considered a challenge to the entire group: