Compared to how this phrase is used normally, the best you can make out of it is "London is to New York what Paris is to its unstated-new-york-to-london counterpart." where this last bit is usually a fairly obvious container class. This weird assumption that the last part of the analogy is one of the first two parts is an attempt to make sense out of something nonsensical, but doesn't strictly follow from the syntactical structure nor colloquial use.

But since there is no obvious new-york-to-london connection, I think the correct takeaway is less that OPs drunk friend has some linguistically-meaningful deep thing to say about London and Paris and more that alcohol is conducive to neither good geography nor good analogies.