The $(x, y)$ plane is the set of all ordered pairs $(x, y)$ of real numbers. The origin is the point $(0, 0)$. The $x$-axis is the set of all points of the form $(x, 0)$, and the $y$-axis is the set of all points of the form $(0, y)$.
I am confused that whether in this definition the term "$(x, y)$ plane" represents an ordered pair, open interval or it is used to just represent that the plane belongs to $x$-axis and $y$-axis.
It refers to the plane, ie, all points of the form $(x,y)$. The notation is the same for an open interval, unfortunately, albeit the context usually serves to disambiguate. Some people use $]a,b[$ to denote an open interval, but that never really caught on. Changing entrenched notation is close to impossible...