For tensors, the following is not distributive:
$$ a_{ij}(x_i + x_j) \ne a_{ij}x_i + a_{ij}y_j$$
However, this is ok:
$$ a_{ij}(x_j + x_j) = a_{ij}x_j + a_{ij}y_j$$
What's the reason for why the first case is invalid, but the second case is ok?
I want to say something like, in the first case, the first term has a free index in slot 1 of "a", but in the second term "a" has a free index in slot2, thus its not distributive?? but, really i have no idea why its not distributive.
I'll assume that you use the Einstein summation convention.
Then the first equation seems to imply that you contract the rank-2 tensor $a=(a_{ij})_{i,j\leq n}$ with the rank-2 tensor $X=(x_i+y_j)_{i,j\leq n}$. However this is something completely different than contracting $a$ with both the vector $x$ and the vector $y$. If this is also what you want to do in the first equation then that is a very strange notation, since inserting the summation symbols (which are implicitly there) would become very hard.
This way, the first equation should be about scalars (all indices are contracted over) and the second one should be about vectors (rank-1 tensors).