I am able to estimate the quantity $$ N(x) = \sum_{\substack{(a,b)=1\\1\leq ab\leq x}}1 = \sum_{n\leq x} \sum_{\substack{(a,b)=1\\ab=n}} 1. $$ For this, I started observing that the inner sum is $2^{\omega(n)}=\sum_{d\mid n} \mu^2(d)$, where $\mu$ is the Moebius function. Then by using standard tools (average order of $\mu^2$, Abel partial summation, ...) I got $N(x) = \frac 6{\pi^2}x\log x + O(x)$.
Unfortunately I have no idea about how to estimate $$ N(x,y) = \sum_{\substack{(a,b)=1\\1\leq ab\leq x\\\frac ab\leq y}}1 = \sum_{n\leq x} \sum_{\substack{(a,b)=1\\ab=n\\\frac ab\leq y}} 1, $$ where $0\leq y\leq 1$. Some ideas:
to derive $\sum_{\substack{(a,b)=1\\ab=n}} 1 = 2^{\omega(n)}$ I noticed that the sum is the number of divisors $a$ of $n$ such that $a$ and $\frac na$ do not share any common factor, which is in turn the number of ways of choosing a subset of the prime distinct factors of $n$;
the two conditions $ab=n$ and $\frac ab\leq y$ implies $a\leq \sqrt{ny}$, so that I should count the divisors of $n$ such that $a$ and $\frac na$ do not share any common factor and such that $a\leq \sqrt{ny}$, but this seems to be difficult to express in terms of subsets of the prime factors of $n$.
I am thus not able to adapt the ideas used to estimate $N(x)$ to something useful for $N(x,y)$. Any help would be appreciated.