I will elaborate on Robin's answer.
There is no unique limit as (x,y) goes to (0,0) --- i.e. the limit does not exist. For instance, if we approach (0,0) on the x axis, we obtain the limit
$\lim\limits_{(x,y) \to (0,0)} \dfrac{x}{x+y} = \lim\limits_{x \to 0} \dfrac{x}{x} = 1,$
and if we approach (0,0) on the y axis, we obtain the limit
$\lim\limits_{(x,y) \to (0,0)} \dfrac{x}{x+y} = \lim\limits_{y \to 0} \dfrac{0}{y} = 0.$
However, we can still consider limits along different curves. The two above limits, along the x and y axes, are two examples: the limit you describe in your question is a similar limit, on the locus of the equation y = mx. This is something different than whether there is "a limit" of f(x,y) as we approach (0,0) --- in this case, because there are different limits depending on how we approach the origin, we may consider the question of what limit one has as one approaches the origin on a particular curve.
In this case, a "curve" is a function
$\big(x(t),y(t)\big) = c(t)$,
for some function $c: \mathbb R \to \mathbb R^2$. "Approaching" (0,0) along the curve c(t) entails specifying some domain for c in which it does not cross itself (for the example of the line y = mx one could take c(t) = (t, mt), in which case the domain can be the real numbers) and for which c(t) = (0,0) for some (unique) value t = T. Then, evaluating the limit of f(x,y) along the curve (x,y) = c(t) means just taking the limit of the composite function f(c(t)) as t approaches T.
In the case of approaching (0,0) on the curve c(t) = (t, mt), we just take T = 0, and evaluate the limit of f(t, mt) as t approaches 0. Up to a change of variables, this is just what you have done above.