As someone who has been a vegetarian for over 25 years I’ve never heard of this. Why would you want your stock to have a thickener? Isn’t it easier to just add a thickener to the food you make so you can control how thick or thin the result is?
You don't add it when you're making or storing the stock. You add it when you're making your sauce. In lots of cases, you're not saucing until the main dish is done and plated.
I am not sure I would use gelatin in a vegetable stock. Things I've used: arrowroot powder, cornstarch, flour, potato starch, agar. A classic veggie gravy would be veggie stock + flour/butter + maybe some tomato paste.
Sorry if I was vague. My question was why you would want your meat stock to have a thickener. I mean I don’t premix all my cornstarch with spices and salt since I want to be able to pick the right amount of aromas and taste separately from how thick I make my soup.
It's more of a mouthfeel thing than a thickener. Gelatin, in the amounts found in a good stock, will still be quite liquid when the stock is at serving temperature, but will sort of coat your mouth with flavor even when the broth hasn't been thickened with a roux or cornstarch.
They were pointing out that with meat stocks it depends what went into the pot - a good home made (or restaurant) one typically already has enough to make it “gel” in the fridge. The box types you mostly get in a store didn’t have this, so might want to add if your recipe assumes it.