Why is The Call of the Wild not just a "dog's story" ?
To me, you can argue that this is not just a "dog's story" because the transformation that Buck goes through is the same as can happen to people in various situations.
What happens to Buck is essentially that he gets taken out of a soft life and thrown into a tough life. Basically, he is told to sink or swim. He must adapt to the new reality or die.
In our lives, we face similar changes (though not usually ones where we literally die if we fail). But when we leave for college we have to adapt to a whole new life. When we get married it's the same. When we get a job, when we have kids, etc.
So the book is at least partly about adapting to change and surviving change. That is a universal subject.
London's The Call of the Wild is a sociological study, so to speak. In fact, it's a little bit didactic (designed to teach), though it is well disguised. In other words, it seems that London set out to make some points about society and philosophy, and created a story about a dog to do it.
Buck changes with his environment, of course, and that is the point. The novel is about the effects one's environment has on one.
The novel is also naturalistic. It features, in this case, a dog, somewhat at the mercy of forces beyond its control.
It can, and often is, read as a simple dog story, of course. But for a sophisticated reader, much more goes on than that.