A lot of small towns I've read about have less than a thousand inhabitants, which is unbelievably small for a "town" in my eyes. In my country of Malaysia in contrast, a village ("kampung") are officially settlements of 10,000 or fewer people -- so a lot of villages here are bigger than some American "towns"!
How did this terminology come to be?
No doubt those clicking on this thread have noticed that it reports 10+ comments but only one, the reminder from automod, remains. (Well, and mine.) The reason most of the other responses had to be removed was because they violated the rule against personal anecdotes. Basically, it was a steady stream of, "I grew up in such and such state and where I lived was called a village" or "I grew up in this other state in a small community but we called it something else."
Taken together, the pattern in responses speak to the answer to your question: They are, but not everywhere, and the context changes based on where you are. It may not seem like something connected to education history, but nearly every time someone asks a question about American schools, I have to offer the same reminder. And that reminder applies here; the authors of the American Constitution laid out what they saw as federal responsibilities but knew they couldn't speak to everything and there were some things they didn't think of or didn't think was a federal responsibility. So, they included the 10th Amendment: The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Basically, naming conventions aren't mentioned in the Constitution, ergo, it's up to each state^1. And since every state has its own constitution, they set the parameters for governing structures. Then, each county in each state has its own charter or constitution which determines naming conventions and finer details around governance. As an example from each coast, New York State has stand-alone villages, villages inside towns, villages that cross town, and villages that cross counties. California is broken up into municipalities and each municipality determines its own name and some of them use village in their name.
This naming diversity did historically create issues around mail delivery, school district creation, and general navigation as the country built a transportation infrastructure. The modern day Zone Improvement Plan (ZIP) code system has served to even out the the linguistic quirks between counties and states. (This response from six years provides a bit more context around ZIP codes.)
^1 And here's another example of the linguistic diversity of American naming habits. Technically, Kentucky, Massachusetts, Pennsylvania, and Virginia are commonwealths but there's no legal difference between a state and a commonwealth. Meanwhile, 48 states are divided into counties but Louisiana has parishes and Alaska has boroughs. Meanwhile, there are some Indigenous Nations or reservations that sit within (or cross) state borders which have their own naming conventions and policies.