When we first meet Steve Rogers, he’s a fiercely independent and patriotic scrawny young man, eager to get into any fight and to defend his country. His transformation into the figurehead for American patriotism through the super serum and the military PR machine is something that we see addressed through subsequent films.
As his attitude to following orders unquestioningly changes, along with his belief in a clear right and wrong, we see his costume change too. The Avengers symbol disappears first, and then the silver star by the time we get to Avengers: Infinity War. However, the real reason for the costume change is merchandising. It’s a lot easier to sell new figures if there’s a change of wardrobe.