According to Wikipedia, asymptote is a straight line. Therefore, neither nor can be an asymptote. Yes, it is possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x) because (the graph of) f(x) is a straight line and g(x) is not. But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.
An asymptote is a straight line by definition.
Edit: It may happen that f(x) is an asymptote to g(x) for some f(x) and g(x). This implies that f(x) is a straight line. This does not imply that g(x) is a straight line; therefore, this does not imply that g(x) is an asymptote to anything.
Well, f(x) is not a family of functions; it's one function. You may consider O(g(x)) as a family, but not f(x). Second, why would you think that f(x) = O(g(x)) is somehow related to f(x) greater than or equal to g(x)? If anything, the first, very rough, idea behind f(x) = O(g(x)) is that f(x) <= g(x). (In reality, f(x) <= C * g(x) for some constant C, and this inequality has to hold only eventually.)