The Blockchain is literally almost exactly like P2P File sharing except the file can be updated across nodes in an extremely limited context.
At a high level, BitTorrent's DHT uses some notion of consensus to communicate between peers about who is on the network and where they and their files are. There is never system-wide coherence of the DHT, but the algorithm (Kademlia) doesn't require anything like that. BitCoin's notion of consensus requires that coherence in all but the last ~10 minutes worth of the blockchain.
Zooming out even farther, BitCoin requires the entire network for anything to happen at all. BitTorrent leverages the entire network to speed up interactions between two individual nodes.
If you're still not convinced, consider what happens if 40% of the network is cut off from the rest for a large period of time. After the extended network partition ends, and the two big swarms rejoin, what do users find is different?
P2P requires seeds and peers. The structure of the Bitcoin network means that as long as there are miners, any information stored on the blockchain will continue to exist. This, in my opinion, makes it much more similar to the idea of a "cloud" than either bittorrent, or a corporate-controlled data server that hosts your information.