Counts for 2-digit primes
Vertical: digit sum (multiples of 3 omitted)
Horizontal: first digit

    |   1   2   3   4   5   6   7   8   9 | total
----+-------------------------------------+------
  2 |   1                                 |    1
  4 |   1       1                         |    2
  5 |       1       1                     |    2
  7 |               1       1             |    2
  8 |   1               1       1         |    3
 10 |   1       1               1         |    3
 11 |       1       1               1     |    3
 13 |                       1             |    1
 14 |                   1                 |    1
 16 |                           1       1 |    2
 17 |                               1     |    1
----+-------------------------------------+------
tot |   4   2   2   3   2   2   3   2   1 |   21

Max count is 3 for digit sum 8