In the following expression
$$ \int_{y_2}^{y_1}\cfrac{\partial P}{\partial y}dy = P(x, y)|^{y_1}_{y_2} $$
I understand algebraically the transition (it is the composite rule and Fundamental Theorem of Calculus for $yy$ direction of the multivariable $P$ function). But what does it mean geometrically? I feel like it is right from a manipulation standpoint but can't quite grasp the geometric meaning of to what it is referring to in a graph. Is it just the projection of the function in the $zOy$ plane?
For an arbitrary but fixed $x$ define the function $f_x$ by $f_x(y) = P(x,y)$. Then, by definition, $$ \frac{\mathrm{d}}{\mathrm{d}y} f_x(y) = \frac{\partial}{\partial y} P(x,y) $$ and so your statement is equivalent to the Fundamental Theorem of Calculus applied to $f_x$: $$ \int_{y_1}^{y_2} \frac{\mathrm{d}}{\mathrm{d}y} f_x(y) = f_x(y) \bigg|_{y_1}^{y_2}, $$ which can be interpreted as a geometric statement about net change of $f_x$ or area under the curve of $(f_x)'$ in the plane at fixed $x$-coordinate parallel to the $yz$-plane, as you suggest.