The physical storage company holds your stuff - and eventually liquidates it - because you're paying to rent that space and your stuff is inside of their physical property. You aren't paying monthly rent to access the app store when you buy an iPhone, that's supposedly free to access (the parent mentions not being able to even update free apps). Apple bricking access to the app store changes that equation, they're proclaiming by doing that that the app store isn't always free to access as a baseline.
Apple is certainly within their rights to go that direction as a business. However, I'm certain that's a very different value proposition than what nearly all iPhone buyers think they're getting when they buy their very expensive phone (ie they expect free, unencumbered access to the app store as a minimum offering whether they buy any other Apple services or not).
You're right, it's a legal lien, but with absolutely none of the consumer protections we would require of any other business.
Using your storage company example, they can't just arbitrarily hold someone's possessions hostage the second they don't pay. There's a legal process that must be followed to place a lien against someone else's property.
The storage company can't do anything until a payment is at least 15 days late. They also have to send a written notice to the customer of their intent to place a lien (in the form of locking the customer out) against the property and notify the customer of the exact date the lien will go into effect. The effective date has to be at least two weeks out from the date of the written notice to the customer.
Would we allow an automobile financing company to repossess a car the day after a missed payment? What about a mortgage company foreclosing on a house? Of course not. But Apple can ostensibly cut off someone's app store access, and their access to free apps and updates to applications they already have a license to use, instantly and without prior notice. Not even the government has that kind of unchecked power over an individual's personal property.