Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$.
Consider a coordinate system in $\mathbb{R}^2$. Every point is represented with an $x$- and $y$-coordinate.
Representing points on the elliptic curve you have to choose $x$- and $y$-coordinates as well, fulfilling the curve equation. So why are elliptic curves points defined over $\mathbb{F}_p$ and not over $\mathbb{F}_p^2$ ?
"A point over $\mathbb{F}_p$" means exactly a point such that both of its coordinates are in $\mathbb{F}_p$, so (as you say) you have a point of $\mathbb{F}_p^2$ (actually, you really have a point in the projective plane over $\mathbb{F}_p$). It's just the way you say it. It is a useful terminology because you can also define points over $\mathbb{F}_p$ in an "abstract" elliptic curve (and other more general objects), which does not come with a chosen embedding into the projective plane (so there is no canonical choice of "coordinates" for the points).