[Fiji Hill]

Since there were more than two states involved, shouldn’t it be: “The War AMONG the States”?

It should be. I believe it was Confederate vice president Alexander Stephens who coined the term "War Between the States.

[Hawthorn]

>> shouldn’t it be: “The War AMONG the States”? <<

No, certainly not. There were only two sides in that war:

1. The United STATES of America, and

2. The Confederate STATES of America.

There was no third grouping of “states” — for example, no “Independent States of America” or no “Federal States of America.”

Therefore, since the war involved just two sides, each of which titled itself as a collection of “states,” then the operative preposition clearly should be “between” — as opposed to “among.”