Physical currencies too have the similar features. Over time everything have well advanced. Bitcoin wallet serves as one's own bank, because it is you who can control it. Banks are different, and OP have mentioned it just to indicate the reason that attracts.
Of course, there will be some similarities between fiats and Bitcoin, but I must say that Bitcoin is more advantageous and makes you feel that you truly own the coin you possess. Fiat has cheated us a lot, and the large difference in usage of the most popular payment system between countries irritates me. But with Bitcoin, the owner has the absolute right and would be able to pay and enjoy being paid as long as the dealer allows the coin.
In other words, with BTC, you can do payments that the best countries in the world could do with the same rights, unlike fiats in which they discriminate.