Consider we have an $n\times n$ square. And for each element $a_{ij}$ there is a $L_{ij}$ set of permissible values(numbers) where $|L_{ij}| = n - 1$. Need to choose a value for each $a_{ij}$ element so that no orthogonal (row or column) contains the same number twice.
Where can I read about such kind of squares? Any books or links?
Or any information about squares where we are choosing value of each element from it's permissible set.
On
Your question can be rephrased in terms of list colourings of the graph with vertices $a_{ij}$ and edges $a_{ij}a_{i'j'}$ whenever $i=i'$ or $j=j'$. If you think about it for a bit, you'll see that this graph is $K_n \square K_n$, the Cartesian product of two complete graphs.
With this in mind, you might find a paper on list colourings of Cartesian products of graphs relevant.
In general that can't be done. Suppose $L_{ij}=\{1,2,\dots,n-1\}$ for all $i,j$. How are you going to choose the values for the top row without repeating a number?
If you meant $|L_{ij}|=n$ instead of $|L_{ij}|=n-1$, look up "Dinitz problem".