According to the problem, the solution is unique, so any triple of values that satisfies the equation provides a solution. We immediately see that $a=b=c=1$ is a solution, hence the angle is 60 degrees.
What count are the ratios between sides. So we can assume that $c=1$.
The equation is symmetric in $a$ and $b$, and we know it's unique, so as $a$ varies, $b$ varies. We can try to see if this infinite family of solutions has an intersection with isoscele triangles, so we put $b=a$ and we solve
finding $a=b=c=1$. So the angle is 60 deg.
Notice that the fun thing is that $k\cdot (1,1,1)$ is not the unique solution. They are indeed infinite. Example, $a=15$, $b=8$, $c=13$.