I did a little experiment. Fix $a=29$, let $b=1,2,\dots,28$. So, you get 28 data points. Well, these points are already extremely regularly distributed. Taking just the first half, $1\le b\le14$, and rearranging the ratios in increasing order, they are (to three decimals) $$.034,.069,.103,.138,.172,.207,.242,.275,.310,.345,.379,.414,.448,.483$$ To three decimals, and modulo round-off errors, these are the numbers $1/29,2/29,\dots,14/29$, which is to say they are about as regularly distributed as possible. The ratios for $15\le b\le28$ are essentially the same numbers - in fact, the ratio for $(a,b)$ seems to be pretty nearly the ratio for $(b-a,b)$.
If what's happening for 29 happens in general, I think it would explain the original histogram.
EDIT: So I think I see what's going on. We're looking at the numbers $$\sqrt{c^2+d^2\over a^2+b^2}$$ But $b$ is very close to $-ac/d$ (since $ac+bd=1$), so these numbers are very close to $$\sqrt{c^2+d^2\over a^2+(ac/d)^2}$$ which simplifies to $|d|/a$. For fixed $a$, as $b$ runs through the units modulo $a$, so does $d$, since $bd\equiv1\pmod a$. So our ratios are as uniformly distributed as the fractions $|d|/a$, which is very.
