To understand why the war was given this name, we have to first realize who actually called it by that name. It was only the American colonists who called the war by this name. To other people, it was generally seen as one part of the Seven Years’ War. This war was a broader war involving many countries and it was fought mostly in Europe. The war in North America was really just one small part of that war.
When the American colonists named this war, they named it for who they were fighting against. The war was not a war between the French and the Indians. Instead, it was a war against the French and the Indians. The American colonists and the British (and their Indian allies) were fighting against the French (and their Indian allies). Because of this, the war came to be known as the French and Indian War.