It gets the name from "rugby football" from the time where both rugby and international football ("soccer") were considered related sports and often shared rules and associations (such as the London Football Association, from which Princeton imported rules to the US, and which eventually started to split as the sports split further).

The change in shape from a round ball to the "handegg" eventually derived from the first American innovation in "rugby football" of the forward pass. Even with the forward pass, the game required kicks into goals for a long while with the "touchdown" a further later innovation (though also influenced by reimporting rugby rules, as it relates to the rugby "try"). Kick offs, punts, field goals, and extra point attempts all still vestigially remain from the rugby origins even as most of the play in between them changed drastically.

American Football is called "football" because it evolved from the "football family". It's like using the term "romance language": Spanish and Italian sound very different today, but they both share roots in Latin. They've also both changed a lot since the days when Latin was a living language.