I'm not sure internet native is the best word usage here. I would say it's a race to find the best, most anonymous most transparent and easiest to use currency for online payments.
I reckon the world would slowly turn into one with 2 payment methods, which make up the majority of transactions across the world. One for online purchases, like items from Amazon or buying in game currency for a game and the other one to be used in real life situations.
Anyway. If there was truly a race as described, Bitcoin would be winning it
At some point I think internet-native is great term there cause bitcoin was the first currency which takes roots from informational technologies and internet. For online payments, every currency is "easiest" and there are more anonymous ones too, for example Monero.
But I have no idea about "internet's own currency". Wait, if you convert bitcoins in fiat via internet too, then what this means? Bitcoin isn't independent, it's price is calculated in USD, it's not separate from it. So I wouldn't say fiat doesn't do this or that thing. As for now bitcoin is like fiat converter.