To get to P&L we first need sensitivities (partial derivatives). For simplicity, let's assume that $V$ only consumes volatility via $g$, that is P&L and Vega P&L are the same thing:
$$ V(x,y) = g(\sigma(x,y)) = g(f(X(x,y), Y(x,y))) = (g\circ f)(X(x,y), Y(x,y))= U(X(x,y), Y(x,y))$$
Note that
$$ V = U \circ (X, Y)$$
$$ \frac{\partial \sigma}{\partial x} = \frac{\partial f}{dX}\frac{\partial X}{\partial x} + \frac{\partial f}{\partial Y}\frac{\partial Y}{\partial x} $$
$$ \frac{\partial \sigma}{\partial y} = \frac{\partial f}{\partial X}\frac{\partial X}{\partial y} + \frac{\partial f}{\partial Y}\frac{\partial Y}{\partial y} $$
$$ \frac{\partial V}{\partial x} = \frac{\partial g}{d\sigma}\frac{\partial \sigma}{\partial x} $$
$$ \frac{\partial V}{\partial y} = \frac{\partial g}{d\sigma}\frac{\partial \sigma}{\partial y} $$
$$ \frac{\partial V}{\partial x} = \frac{\partial U}{dX}\frac{\partial X}{\partial x} + \frac{\partial U}{\partial Y}\frac{\partial Y}{\partial x} $$
$$ \frac{\partial V}{\partial y} = \frac{\partial U}{dX}\frac{\partial X}{\partial y} + \frac{\partial U}{\partial Y}\frac{\partial Y}{\partial y} $$
In the P&L context, the Taylor expansion/approximation is used:
$$ V(x,y) - V(x_0,y_0) = \frac{\partial V}{\partial x}(x_0,y_0) (x-x_0) + \frac{\partial V}{\partial y}(x_0,y_0) (y-y_0) $$
$$ + 0.5 \frac{\partial^2 V}{\partial x^2}(x_0,y_0) (x-x_0)^2 + 0.5 \frac{\partial^2 V}{\partial y^2}(x_0,y_0) (y-y_0)^2 $$
$$+ 0.5 \frac{\partial^2 V}{\partial x \partial y}(x_0,y_0) (x-x_0)(y-y_0) + 0.5 \frac{\partial^2 V}{\partial y \partial x}(x_0,y_0) (x-x_0)(y-y_0)$$
$$+ ..., $$
which can be notationally abbreviated as
$$ dV = \frac{\partial V}{\partial x}dx + \frac{\partial V}{\partial y}dy + 0.5 \frac{\partial^2 V}{\partial x^2}(dx)^2 + 0.5 \frac{\partial^2 V}{\partial y^2}(dy)^2
+ 0.5 \frac{\partial^2 V}{\partial x \partial y}dxdy + 0.5 \frac{\partial^2 V}{\partial y \partial x}dx dy + ...$$
If one drops the second-order terms, we get the usual:
$$ dV = \frac{\partial V}{\partial x}dx + \frac{\partial V}{\partial y}dy $$
Note that $$ \frac{\partial V}{\partial X}, \; \frac{\partial V}{\partial Y} $$ can be ambiguous notations.
Pricing function $U$ (different from $V$) whose 'inputs' are $X$ and $Y$, as directly observed numbers, has its own P&L:
$$ dU = \frac{\partial U}{\partial X}dX + \frac{\partial U}{\partial Y}dY, $$
that is
$$ U(X,Y) - U(X_0,Y_0) = \frac{\partial U}{\partial X}(X_0,Y_0) (X-X_0) + \frac{\partial U}{\partial Y}(X_0,Y_0) (Y-Y_0) $$
If we want to involve $U$'s sensitivities in $V$'s P&L calculation, we use (from above list of chain rule applications):
$$ V(x,y) - V(x_0, y_0) = \left(\frac{\partial U}{dX}(X_0,Y_0)\frac{\partial X}{\partial x}(x_0, y_0) + \frac{\partial U}{\partial Y}(X_0,Y_0)\frac{\partial Y}{\partial x}(x_0, y_0)\right) (x-x_0) + \left(\frac{\partial U}{dX}(X_0,Y_0)\frac{\partial X}{\partial y}(x_0, y_0) + \frac{\partial U}{\partial Y}(X_0,Y_0)\frac{\partial Y}{\partial y}(x_0, y_0)\right)(y-y_0), $$
where $X_0 = X(x_0,y_0), Y_0=Y(x_0,y_0)$.
If we want to involve $V$'s sensitivities in $U$'s P&L calculation, we use (assuming invertibility):
$$
\begin{bmatrix}
\frac{\partial U}{dX} \\ \frac{\partial U}{dY}
\end{bmatrix}(X_0,Y_0) =
\begin{bmatrix}
\frac{\partial X}{dx} & \frac{\partial X}{dy} \\
\frac{\partial Y}{dx} & \frac{\partial Y}{dy}
\end{bmatrix}^{-1}(x_0,y_0) \cdot
\begin{bmatrix}
\frac{\partial V}{dx} \\ \frac{\partial V}{dy}
\end{bmatrix}(x_0,y_0)
$$