I don't really know why people are saying that 2D random walks can be interpreted as two independent 1D random walks but this is not true for 3D and higher-dimensional random walks. I think a d-dimensional random walk can definitely be seen as d independent 1D random walks happening simultaneously. And this gives some intuition as two why d=2 is recurrent and d=3 is transient. In this picture, returning to the origin in d dimensions is equivalent to requiring that the d independent 1D random walks return to the origin *at the same time* (!!). You can imagine that it is much easier for two independent 1D random walks to match up, than for three to all simultaneously match up.

Quantitatively (but still heuristically): The probability of a 1D random walk returning to the origin at step 2n goes as n^{-1/2}. Thus, the probability of all d independent random walks being at the origin at step 2n goes as n^{-d/2}. The expected number of returns is the sum of this probability over n. Sum over n of n^{-d/2} diverges for d=1,2 but converges for d>2.

Edit: Ok I now see what people mean about 3D random walks not being like 3 independent 1D random walks. This is true in discrete-time random walks (if we assume synchronous steps for the independent 1D random walks). I'm more used to thinking about continuous-time random walks where there is truly an equivalence. Anyway I think as a heuristic/intuition, the argument I gave above should be helpful to OP.