Is this the correct "standard form" of nonlinear programming (optimization) problem and if it is why it's in this form?
There is no "standard form" that you must use. You don't have to write an optimization problem in that way. This is a useful representation in a number of ways; it is very general; and it facilitates some kinds of analysis (e.g., KKT conditions).  But there are no police forcing you to express your optimization problem in this way; you can use whatever you find most useful.

Note that every optimization problem can be expressed in this way, as $h(x)=0$ can express arbitrary constraints on $x$ and $f(x)$ can express an objective function.  So, it is nice from this perspective.