It's hard to test without having code I can run, but I changed a couple lines (marked w/ comments in the code and described more completely below) to get the functionality I think you want. Does the following work how you want it to?

sub extractData
{
    my @projects;

    push( @projects, [readOnly("$proj Data.txt"), "A"] );

    unless( $pointerProj eq "NONE" )
    {
        push( @projects, [readOnly("$pointerProj Data.txt"), "B"] );
    }

    for( @projects )
    {
        my @data = @{ $_->[0] };    # WAS: my @data = @{ $_[0] };
        my $key = $_->[1];          # WAS: my $key = $_[1];
        ...
    }
}

sub readOnly
{
    my @contents;
    my $filePath; my $file;
    my $errMsg;

    $filePath = "$path\\" . $_[0];
    $errMsg = "Unable to read $filePath: $!";

    open $file, '<', $filePath or die $errMsg;

    @contents = <$file>;

    close $file;

    return \@contents;              # WAS: return \@contents;
}
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I changed readOnly() to return an array reference. Each project is then an array reference with another array reference (the 'data') at index 0 and a text string (the 'key') at index 1. To get these two items, you need to dereference the outer array reference with $_->[X]. You used $_[X] which is how you get an item from an array, not an array reference.