The situation gets a bit more complex with Lightning Network, as there we do have a peer-to-peer networking, but unlike Bitcoin, we don't have a global state or consensus other than nodes gossiping about the structure of that peer-to-peer network.
Lightning Network is composed of 2 others: Testnet and Mainnet, which are not linked to any blockchain (not even Bitcoin one). Lightning is a multiasset protocol and today nothing prevents you from adding Bitcoin to the Lightning mainnet or adding Bitcoin Testnet coins there. So, when we say 'network' in terms of Lightning mainnet or testnet, we don't imply a particular blockchain it's based on. Of course, usually when speaking about Lightning mainnet working on a blockchain we mean Bictoin mainnet, but technically there are no limitations to use any other one as a base layer, including the testnet one or even a fork of Bitcoin (e.g. Litecoin). What is reflected in Lightning Network by words 'testnet' or 'mainnet' is the fact whether the creators of the network consider the functionality stable/final or not. Testnet was, in fact, the version that was not adviced to be used for real payments and mainnet was said to still be in beta (while from Bitcoin point of view, if you are reckless enough, you could perform real-value payments already). And this situation is pretty natural for Lightning, as again, it's a multiasset protocol and it's about peer-to-peer networking, thus all the funds are managed by the channel peers, bidirectionally, inside a single channel, therefore all you are potentially risking to loose is the amount of funds in this channel.