Linear Definition is as follows $$L(x + y )= L(x) + L(y) $$ $$L(ax) = aL(x)$$
I get confused with the definition for multivariate linear function. Let's say we have a function like below. $$L(x, y)= x^2 + yx^3$$ Can I say this function is linear with respect to $y$? If so, how the definition for linear function is used here, and or not why this is not a linear function with respect to $y$?
(If you give me a useful link for this, then I really appreciate it.)
That's not what the linearity means. First, you can't say that your function is linear with respect to $y$ because $$L(x, \lambda y) = x^2 + \lambda yx^3 \neq \lambda (x^2 + y x^3) = \lambda L(x,y),$$ which is to say it fails one of the conditions.
In your case, linearity means that for all $(x_1, y_1)$, $(x_2,y_2)$ you have $$L(x_1+x_2, y_1+y_2) = L(x_1,y_1) + L(x_2,y_2).$$ In other words, you take the sum of the components to the sum of the images for all coordinates, not just one of them.
Multiplication by a scalar, which is your second condition, means that you'd have to multiply both $x$ and $y$, that is, $$L(\lambda x, \lambda y) = \lambda L(x,y).$$ Wikipedia has a nice set of examples, in particular, the last one with gifs.