It seems they mean grocery store, rather than supermarket. Also take into account that in the Netherlands it's more common to go grocery shopping daily or every few days.
In the Netherlands you have supermarkets, some big, some small and sometimes smaller mom and pop stores that sell the basics. They are outcompeted by supermarkets that tend to be very close as well. There are no supermarkets that sell everything, like you might find in France (Hyper Marche, Carrefour) or the US (Walmart).
The person you're responding to mentioned them as "minimarkets which are really small". They still exist and are very important as a last resort thing, or for exotic stuff.
In France sometimes they're called "arab stores" or something along those lines, because they're often held by people of visibly foreign origin (tbf I've seen more Indian/Pakistani/Sri Lankan/Bangladeshi than Arab-held ones), and are great for emergency shopping on Sunday/late at night when you realise you're missing something, but also for exotic products you wouldn't find in a regular supermarket like exotic spices or even high quantities of rice.
They still exist, you just need to know where to find them.
Your corner bakery, butcher & vegetable store are still around, but they tend to serve an upmarket / luxury customer segment these days.
In most big(ger) cities there's "night stores", often run by people of Arab origin (like Lebanese), "toko" (Indonesian grocery store), and Turkish supermarkets (my go-to for dried legumes like lentils, chickpeas, mung beans etc).
Relatively new are organic / "bio" focussed supermarkets. Usually expensive... But they sell many products that regular supermarket chains don't.
That means "things required for daily life". They definitely mean supermarket. It's about the same in Rotterdam. I'm an eight minute walk away from four different big supermarket chains. If I walk 10 minutes I can get to the second most distant Albert Heijn.