in a distributed system, conflicts are resolved by majority vote, and voting has to be made expensive to prevent stuffing the ballot-box. Proof-of-work is one way to make it expensive. Proof-of-stake is another.
I have read the thread, and there was a convincing differentiation of shares vs. coins.
With shares, you vote without working, according to how much you have at stake from previous investment.
With coins, you must pay for every vote you want.
The first one is scarcely suitable to be a basis for a monetary system, because there is no anchor, no cost that "keeps people honest" like in gold standard vs. fiat standard.
Generally with PoS you pay with "coin-days". The longer you hold coins, the more weight they carry for voting, but the act of using them to vote resets their age to zero. So that's what you lose; not the actual coins, but the coin-days. But you do lose something, something that is unavoidable expensive to acquire (you have to buy the coins and then hold them for a time). So it is not like voting with company shares. As you say, that is essentially free once you have the shares; you can vote the same shares multiple times.