Yes, as a (lapsed) amateur butter maker, I'd say this is the main difference between European butter and US butter, not the fat content. If you make your own butter with ordinary cream, it'll taste bland like US butter, irrespective of the fat content. You need to add a culture. Not just any culture, which you might get if you leave your cream out of the fridge long enough before starting, but one of the right cultures. Search for "cultured butter" on the interweb for further information.
As an aside, my personal view is that making your own butter doesn't provide anywhere near the same return on investment as making your own bread - home made butter takes ages, doesn't really taste that much better, and can cost more money, while home made bread doesn't take that long, and can taste a lot better.