Only if the angle is a right angle.
Let $(u,v)$ be the standard Cartesian coordinate system on the plane, so the first fundamental form is (as a line element) $ \mathrm{d}s^2 = \mathrm{d}u^2 + \mathrm{d}v^2 $
Let $y = v$ and $x = u+v$. Clearly the level curves of $x$ and the level curves of $y$ always intersect at a 45 degree angle. But using that $u = x - y$ we get that $\mathrm{d}u = \mathrm{d}x -\mathrm{d}y$ while $\mathrm{d}v = \mathrm{d}y$. So $ \mathrm{d}s^2 = \left(\mathrm{d}x - \mathrm{d}y\right)^2 + \mathrm{d}y^2 = \mathrm{d}x^2 + 2\mathrm{d}y^2 - 2 \mathrm{d}x\mathrm{d}y$
Which means that in this $(x,y)$ coordinates, we have that $E = 1$, $F = -1$ and $G = 2$.
To put it in other words, $F = 0$ is equivalent to the coordinate system being such that the coordinate curves are everywhere mutually orthogonal. This condition is a bit more stringent then just having the same angle everywhere.