I've asked here before, "What is liquidity?", if not literally then sort of metaphorically or something. One of the problems is that the term "liquidity" is used to refer to a few different related but quite distinct things. When we refer to a corporate (especially bank) balance sheet, we're usually referring to the exposure of a bank to sudden, unexpected obligations and its ability to handle them. In particular, suppose our bank has, by any reasonable estimation, an ability in the long run to meet all of its liabilities, but has maybe $50 in cash on hand, and you have a few hundred dollars in an account and the contractual right to go up and try to withdraw $100. This bank's liquidity would be characterized as "Not good." On some level, a company that has a $100 loan maturing today and expects to be able to roll it over (i.e. borrow the $100 from someone else to pay the old loan) is in the same position; perhaps in a well-functioning financial market they should be able to do that, and are thus in the long-run "solvent", but if something goes awry they find themselves in breach of some obligation.
So let's compare Matt Levine's description of Lending Club; it appears that Lending Club, with (by the relevant accounting rules) a very high leverage ratio, has little if any liquidity exposure; its contracts are written in such a way that it only owes a creditor money as long as a debtor pays it that money. While this may, as he says in footnote 6, leave them with a business risk, it leaves them without a risk that they will suddenly have to meet the sort of legal contractual obligation that a bank intermediating lending would have.
Five to ten years ago — and maybe today, but I'm less well-connected now — a lot of hedge funds were making money on liquidity premiums, deliberately buying things like wind farms that were somewhat likely to be profitable but couldn't be sold as quickly as (say) IBM stock if a sudden need for cash arose. Wind farms are actually a specific example I remember being presented to me of an illiquid asset that was owned by a financial company; if that company had a contractual obligation (possibly requiring a creditor to trigger it, e.g. by showing up at a teller window) to provide dollars on a moment's notice, the wind-farm may be a good asset for the long-run but it won't provide liquidity protection from that event. If the company had a contractual obligation to provide, say, good title to a wind farm at a moment's notice, all of a sudden this looks like a great asset from a liquidity standpoint; better (at least from the standpoint of that (weird) risk) than cash.
That isn't meant to be entirely* fatuous; certainly traders of derivatives for physical delivery (e.g. oil futures) occasionally find themselves in analogous positions. If the world foreign exchange market were in turmoil, a dollar balance might be useless to fund an obligation in rubles, even if there were some objective sense in which, at any reasonable exchange rate, it would be high enough to cover the debt. What it does mean is that "funding liquidity", which is more or less what we're discussing, is about matching assets and liabilities in a way that is much more general than maturity matching in general, and that to the extent that it is, in practice, something that can be captured usefully in a low-dimensional way, that fact is phenomenological and not theoretically fundamental.
* Only mostly.