If I write $x \approx y$, does this mean (a) $x$ is sufficiently close to $y$ for some practical purpose, or (b) $x$ is sufficiently close to $y$ for some practical purpose, but is not equal to $y$?
If (a) is true, then it appears $x \approx x$.
This question appears to have more importance when considering something like the small angle approximations. Is the statement $\sin(x) \approx x $ true when $0 \leq x \leq 1$, or true when $0<\sin(x) \leq 1$?
Keeping in mind that "$\approx$" isn't actually a formal notion at the outset, any reasonable approach to it (e.g. via nonstandard analysis, where "$\approx$" is interpreted as "is in the halo of") will say that it does extend equality: everything is approximately equal to itself.
It's worth noting that the way approximation is cropping up here is in questions of asymptotics: when we say "$x\approx \sin(x)$ near $0$," we don't really care about how close $\sin(0.1)$ is to $0.1$, what we're really interested in is $\lim_{x\rightarrow 0}{\sin(x)\over x}$. There is a formal framework for discussing this sort of analysis, namely big-O notation and its relatives, and in this framework everything is indeed close to itself.