It’s important to distinguish the function $F:\mathbb R^2\to\mathbb R$ defined as $F: (x,y)\mapsto x^2+y^2-6xy$ from the equation $F(x,y)=0$. For this function, $x$ and $y$ are independent variables—they can take any values whatsoever—so it makes perfect sense to compute its partial derivatives with respect to each variable by treating the other variable as a constant. The graph of this function is some surface in $\mathbb R^3$ and at points at which it’s differentiable (everywhere for this function), this surface has a tangent plane with normal vector $(F_x,F_y,-1)$.
The equation $F(x,y)=0$, describes a level curve of the function $F$, specifically the intersection of its graph with the $x$-$y$ plane. The tangent line to this curve at a point is just the intersection of the tangent plane to the graph of $F$ with the $x$-$y$ plane. The projection of the tangent plane’s normal onto the $x$-$y$ plane, namely $(F_x,F_y)=\nabla F$ is orthogonal to this tangent line. The slope of this vector is $m_1 = F_y/F_x$, and using the relation $m_1m_2=-1$ for the slopes of perpendicular lines, we see that the slope of the tangent line is $m_2=-F_x/F_y$.
To put it another way, when you compute $dy/dx$ via partial derivatives of $F$, you are viewing the curve implicitly defined by the equation $F(x,y)=0$ as a horizontal slice through the 3-D graph of $F$ and then finding a tangent line to this curve by taking the same slice through the tangent plane to the graph of $F$ at that point. By adding an extra dimension and stepping out of the $x$-$y$ plane in this way, you can temporarily ignore the fact that $x$ and $y$ are not independent for points on the level curve. This dependency doesn’t disappear completely, though. It still plays a role in computing the coordinates of a point on the level curve, which you need in order to compute the partial derivatives of $F$ there.
An important thing to note about the above approach is that nowhere does it require that there exists some function $f:\mathbb R\to\mathbb R$ such that $F(x,y)=0$ implies that $f(x)=y$. Indeed, there may not even be such a single function. E.g., $x^2+y^2=1$ can’t be the graph of any function since it fails the vertical line test. (It might still be possible to find such a function in the neighborhood of a point, though—that’s what the implicit function theorem is about.) When you perform implicit differentiation, you start off by assuming that there is such a function and then differentiate both sides of the equation $F(x,y)=0$ taking that into account.
These two approaches can be reconciled via the chain rule. Suppose that $y=f(x)$. To help keep things straight, we introduce the function $\phi:\mathbb R\to\mathbb R^2$ with $\phi:x\mapsto(x,f(x))$, and also $g=F\circ\phi$. Then $$g' = \nabla F\cdot\phi' = (F_x,F_y)\cdot(1,f') = F_x+F_yf'.$$ This expression takes into account the assumed dependence of $y$ on $x$, but it nevertheless involves partial derivatives of $F$. Since we have $F(x,y)=g(x)$, implicit differentiation is equivalent to differentiating both sides of the equation $g(x)=0$ and then solving for $f'$, but this gives $F_x+F_yf'=0$, from which $f'=-F_x/F_y$ as before.
