Now, suppose the transformation(in 2d) I am working with has two
separate functions
for $x$ and $y$.
That is, the transformation for $x$ is of the form
$$
x'=\frac{x}{x+y}
$$
and the transformation of $y$ is
$$
y'=\frac{y}{x+y}
$$
Each is an LFT (The schwarzian derivatives are $0$) but is
the transform as a whole still considered an LFT?
I was under the impression that the term "fractional linear transformation" referred to transformations in one (generally complex) variable. However, it is certainly still valid to extend this idea to more variables; you get elements of the projective special linear groups.
I don't think this is really the concept you're looking for, though, since your transformations have an extra property (they are homogeneous).