In unregulated environments, sure. I mean, Bitcoin did spawn things like Monero and Ethereum, among other popular crypto that aren't just clones. The competition aspect is more tech than finance though, so there's that.

Once something tries to kill fiat in a survival of the fittest scenario, governments are essentially judge, jury, and executioner. Competitors are even dead right out the gate in some territories, where their official currency is the only legal tender by law. It's also why companies like Paypal are competing for a way to make spending fiat easier, because they can't touch fiat itself (remember Libra? lol).

I guess fiat vs. fiat is also a thing, and China has been trying to beat the USD as a reserve currency, but nobody but the countries involved themselves actually benefit from that, so maybe the general answer is no?