Nice question. If I programmed it right one cannot quite get 7.
There are dozens of ways to put 7 "superqueens" (a.k.a.
on the board so that there's only one attacking pair, e.g.
a5,b3,d6,e7,f1,g4,h8 with one B-attack or
a2,c7,d4,e1,f8,g5,h3 with one N-attack.
I looked only for configurations with no R-attacks;
such a configuration can be represented
by a permutation of the numbers from 1 to 8 with one number missing.
None of the one-off configurations fits into a $7 \times 8$ board,
though it might be possible to do that with two "superqueens"
on the same rank or file and no other attacking pairs.
Here is the gp code. "k" is the empty column,
which can be assumed to be 1, 2, 3, or 4 by symmetry.
Q(x1,y1,x2,y2) = ((x1-x2)^2 + (y1-y2)^2 == 5) || (abs(x1-x2) == abs(y1-y2))
\\ true iff (x1,y1) is a N- or B-move from (x2,y2)
a = numtoperm(8,n);
s = sum(i=1,7,sum(j=i+1,8, (i!=k) && (j!=k) && Q(i,a[i],j,a[j])));
The output is empty. Changing "s==0" to "s==1" yields 172 solutions, including
2[5, 2, 3, 6, 7, 1, 4, 8]
2[2, 6, 7, 4, 1, 8, 5, 3]
which correspond to the B- and N-attack examples respectively.
P.S. Extending this calculation to a $9 \times 9$ board finds that
9 "superqueens" still cannot be accommodated, but there's
a unique permutation (up to the 8 board symmetries)
that fails at only one pair. I do not exhibit this
configuration here because finding it might make for
a fun puzzle. There's a number of nonattacking configurations
of 8 superqueens on this board; alas none has any symmetry, nor does any fit in an
$8 \times 9$ rectangle $-$ indeed only one (up to symmetry)
omits the second column. I leave this last as a puzzle as well.