If you need unique characters from Array A - i.e. it's similar to a pack of cards, you can only have one of each - then you need to remove each character from the input when you use it.
The simplest way to do that is to convert your array to a List and use RemoveAt each time:
List<string> AB_Concat = new List<string>();
List<char> data = A.ToList();
Random random = new Random();
for (int i = 0; i < 26; i++)
{
int index = random.Next(data.Count);
AB_Concat.Add(Convert.ToString(data[index]) + Convert.ToString(B[i]));
data.RemoveAt(index);
}
for (int i = 0; i < 26; i++)
{
Console.Write(AB_Concat[i] + " ");
}