As far as I can see, the point of the question is that the mathematical definition of "mutually independent" includes some situations that don't fit your intuition of what "independent" should mean. Such things happen fairly often in other parts of mathematics too, because we use ordinary English words that might carry some unwanted connotations. An extreme example is that, in ancient times (perhaps even in medieval times --- I'm not sure about that), "number" meant what we now call "natural number greater than or equal to 2". The idea that 1 is a number still clashes with some people's intuition; if I say "a number of people think I'm a genius" and it turns out that only one person (guess who!) thinks so, was I lying? The idea that 0 is a number is, for some people, even more counter-intuitive. Nevertheless, broader notions of "number", including not only 1 and 0 but negative numbers, real numbers, and complex numbers, have been fruitful enough that mathematicians adopt these meanings and discard the old, more intuitive meaning.
In the case at hand, something similar may be at work; the official definition is (at least) easier to work with than a variant that requires at least two random variables. For example, we can say that, if a set of random variables is mutually independent, then so is any subset.
In addition, though, I think it is possible to explain the official definition in a way that makes intuitive sense. Consider what dependence (the opposite of mutual independence) of a family of random variables should mean intuitively. To me, it means that, if we're given the values of some of the random variables in the family, then this information influences the distribution of some other random variable in the family. Now this will never happen when the family consists of only one random variable. Since we can't have (my intuitive notion of) dependence in this case, it makes sense to declare any family of just one random variable to be mutually independent. (For the same reason, I would consider the empty family of random variables to be mutually independent.)