In this declaration
std::array<std::array<int,2>,2> x = {{0,1},{2,3}};
you have three nested aggregates. The first pair of values enclosed in braces
{0,1}
is considered by the compiler as an initializer of the second aggregate that is present in the declaration as one sub-aggregate. So the second pair of values in braces
{2,3}
are considered by the compiler as redundant that has no corresponding object.
You could declare the array for example like
std::array<std::array<int, 2>, 2> x = { { {0,1},{2,3} } };
The braces may be elided when an aggregate is initialized. (C++17 Standard, 11.6.1 Aggregates)
12 Braces can be elided in an initializer-list as follows. If the
initializer-list begins with a left brace, then the succeeding
comma-separated list of initializer-clauses initializes the elements
of a subaggregate; it is erroneous for there to be more
initializer-clauses than elements. If, however, the initializer-list
for a subaggregate does not begin with a left brace, then only enough
initializer-clauses from the list are taken to initialize the elements
of the subaggregate; any remaining initializer-clauses are left to
initialize the next element of the aggregate of which the current
subaggregate is an element.
So in this declaration
std::array<std::array<int,2>,2> x = {0,1,2,3};
the braces are elided and the aggregate is initialized as it is described in the quote..
In this declaration
std::vector<std::vector<int>> y = {{0,1},{2,3}};
there is used the constructor of the class std::vector that accepts std::initializer_list as an argument. In this case the constructor builds as many elements of the vector as there are elements in the initializer list.