The difference between "the way" and "like"

Solution 1:

Using the way describes the more concrete and specific chain of events in the process of treatment. If you say that someone treats their children the way an officer treats his soldiers, you're basically saying the whole process is the same.

Using like means that they behave towards their children as if they were soldiers, but don't necessarily force them to go through the same routine as soldiers do.

Imagine and compare the following:

My father treats me like a child, although I'm not a child anymore.

This means he imagines I'm still a child and behaves as if I were one towards me, but that doesn't necessarily say what exactly he makes me do.

My mother is happy with how she raised my older brother, so she treats me the way she treated him.

This means she makes me go through my childhood exactly the same way she did with my older brother, as she's satisfied with how he's been raised.

Solution 2:

"Like" here is used as an informal "as."

In short,

He treated his children as an officer treats his soldiers.

He treated his children (in) the (same) way (as) an officer treats his soldiers.

He treated his children like an officer treats his soldiers. (informal spoken English)

All the sentences above are correct.