Actually it's fair to say most of the business is done by local chains, and Europe-wide operations are rather uncommon.
From your mentions, I'd only consider to be "Europe-wide" Carrefour and Spar. Carrefour is everywhere (though their market strategy changes a lot from country to country) and Spar's operations are kinda spotty but also kinda everywhere.
Auchan is everywhere, but in heavy decline because they're basically the Walmart (supercenter) of Europe and people don't like to drive to a hypermarket that much anymore. Almost their only profitable branch at the moment is their Chinese one.
Lidl and Aldi's presence is very spotty outside of Germany. Many countries copied their hard discount model so they have to fight with local companies that have a cultural edge (Lidl and Aldi tend to have tons of German products no matter the host country).
Tesco is a leader in the UK, then kinda common in Central-Eastern Europe, but then almost universally unknown in Germany-Austria-Italy and westward. There are more Tesco in Thailand than in the whole of Continental Europe, to put it in perspective.
Rewe is just in Germany, Austria and Eastern Europe (under different names and with wildly different marketing).
Iceland I never heard about, but it's overwhelmingly a UK operation from a quick search.
From your mentions I'd take the wild guess that you live (or have lived) a significant amount of time in the UK ;)