The point of insurance is to decrease volatility. Stock options do the opposite of this. I don't understand how this comparison makes any sense.
For example if I have 100 shares of AAPL @108 and I'm nervous of it dropping below $80 in the near future, I may buy a MAR16 put option at $90/share for a few pennies in the hopes that if the stock does actually decrease to a level that makes me feel uncomfortable, the options will increase enough to offset the stock loses.
So they very much can be insurance. It just depends on how you play them.
The comment to which you replied was talking about employee stock options, which are not generally used to hedge a short position, and often represent a large part of someone's overall wealth and/or income.
You are of course correct that stock options on the public market are most often used to hedge and thus they do act like insurance. It seems fairly certain that this is not what OP had in mind, though.
If a couple buys "trip insurance", it's more about managing risk and uncertainty so they can sleep at night if their cruise ship is cancelled rather than smoothing over price volatility.
Also, I'd just read the parent's quote about "insurance" as a rhetorical device[1] rather than some rigorous financial instrument. The author thought of SOs as "insurance" against missing out on potential upside.
[1] both "insurance" and "lottery ticket" to describe stock options are analogies/metaphors
To rephrase, they're definitely paying a small amount of money to avoid maybe losing a large amount of money. That sounds like the definition of decreasing volatility to me.