Here's an image from bricklink for the catalog item 15573:
It seems clear to me that 15573 and 3794 are different parts.
Not distinguishing between them I would consider lazy, if not bad practice.
Wouldn't bother starting an argument with that store, as they basically did warn you, but I'd blacklist the store for future purchases.
There are various cases and some shops tend to have quite relaxed rules for what they sell in terms of part variation or used brick policy (basically how much brick is used to be considered acceptable for sale by that particular shop).
In your particular case the warning you mentioned was stated in that particular shop's Terms & Conditions. You had to ...
I'd say this is something to look out for when dealing with a part that BrinkLink itself notes is very similar to other part numbers. Reading the detailed notes about buying sets is more important than parts generally. I'd read a bit more closely about the stated condition of used parts than new parts.
I've made a few dozen orders on BrickLink over the ...