Actually, money is no exception. We have dollars, euros, pounds, pesos or whatever, and they are all competing with each other at an international scale (or, rather, economics are). You can hold your savings in whatever currency you see most appropriate (or profitable). Also, If I'm not mistaken, bitcoin is not considered illegal in the USA, though it is obviously not a legal tender there.
Crypto currencies are the solution to fiat! BTC was built with a limit of 21M units because Sastoshi knew the evils of inflation. The distributed nature of the system means that it essentially cannot be shutdown. Governments accepting Bitcoin is not because they wish for competition, but more because they don't have a choice. Of course most of them at this point are ignorant of the technology and will be pressured in due to demand as well.
Actually, cryptocurrencies are not very far from fiat. Both bitcoin and fiat are money since they are called so and accepted as such (no utility besides being a currency). Regarding the limit of 21M coins, it is not really an advantage since, in the first place, it won't be reached any time soon, and it can be changed afterwards if such a necessity should arise, in the second.
As to Bitcoin's generation rate, it is also a double-edged sword (there are pros and cons to this).
I'm talking about the essence of bitcoin, not about formal definitions (sorry if it wasn't evident from the start). To keep things simple, think that Satoshi Nakamoto issued an edict that made bitcoin into a money.
And I didn't understand what you meant by "fiat has nothing to do with its utility".