There is a well known game called "Clash of Clans", and its title's meaning is obvious, while "Clan Clash" may mean "Clash within Clan" (or just does sound naturally perhaps?), and "Clans Clash" is not correct either, as far as I understand. But what about the case, that I mentioned in title? Since there's not such ambiguity, can I say "dog clash", or "dogs clash" to express "clash between dogs"?
While "Clan Clash" could be interpreted to mean a clash within a clan, I think it's more likely to be interpreted as a clash between two rival clans.
As for a clash between canines, "dog clash" would be the way to say that. (We typically use the singular, even though there are more than one involved.) For example:
A cola clash would likely be regarded as a "war" between two cola companies (in the US, that would be Coca-Cola and Pepsi (as was used in a Washington Post headline: Cola Clash Is Taking To Heavens.)
A burger clash might be a clash between two or more hamburger chains or restaurants. For example, a 2018 food column was entitled Burger Clash - A battle of the tastiest in Korea (about the search for the best hamburger one might find in Korea).
A cat clash would be a clash between two or more felines; it's often used in sports headlines to describe a contest between two teams with names such as the Tigers vs the Jaguars, or the Leopards against the Lions.