Do you also consider Google and Bing on the same footing?[0] No, right?. But in anycase, fair enough, the technical correct one is duopoly—the problem just is, it doesn't make it any better.
Oligopoly. Discover card and American Express are both mostly accepted (I say mostly because some companies don't accept American Express due to their charge back policies).
And yes, it makes it so much better. True monopolies can slack so hard and yet still rake in boatloads of cash because their customers have no other option. Google cannot sit around on their search engine or other companies will steal all their customers. Just look at how duck duck go is growing in popularity.
And yet here we are, living in a world where PayPal, visa, and MasterCard are successful able to censor lots and lots of people.
So, by definition, this is a problem.
We would no longer have a problem if we get to the point where a single company censorsing your transactions is no big deal because a dozen others will immediately step up in their place.
But we don't live in that world, and so because of this we have to recognize that there is indeed an oligopoly problem.
Does that matter in Europe? From what I have read here, most people use debit cards (interest rates for credit don't matter) and the fees are capped super low so the % fees don't matter.
I'm not entirely sure what your question is? If it matters that Discover and Amex is not being accepted?
Most adults will be using credit cards, unless you haven't proven to your bank yet that you are trustworhty (i.e. just moved to the country). Debit is usually younger people, below 18 (at least in DK). Both are still Visa or MasterCard.
It would still be a problem, if as a merchant got banned by Visa/MasterCard, since now they have no other way than to accept cash, which obviously doesn't work if you are an online shop.
This really depends on the country. In Germany, most day-to-day payments happen with cash or over the Girocard network. I got by for most of my adult life without a credit card. The only thing that's kind of tricky is booking a hotel without one (international hotel chains have standardized their registration systems around credit cards). I only got a credit card last year because it's free with my current checking account.
Sometimes people say "monopoly" when they mean "cartel". Are you arguing that these companies do not have substantial market power compared to more competitive markets or are you just trying to distract from the discussion with an argument over definitions? Do we see Mastercard and Visa competing on price (e.g. interchange rates)?
Having 4 main companies (mastercard, visa, discover, and american express) doesn't make you a monopoly or cartel.
Obviously, they do have more market power than purely competitive environments but they do compete a lot on rewards which cut into the profitability of those interchange rates. I cannot find any source to measure the interchange fees - rewards profitability but I did find one that shows most of their profits don't come from those but instead interest rates [0] and I'd bet they don't make so much after the rewards (I get minimum 1% cash back on my cards and sometimes up to 5%).
Rewards are entirely paid for by higher interchange rates. If a card is paying a 1% reward of some sort, it is also costing the merchant at least 1% more than non-reward cards at the interchange level. A lot of this is hidden by services like Stripe and Square that have flat fees for merchants (which are higher than any interchange fee) and also by processing ISOs that hide the interchange rates in their own synthetic flat fees. Nevertheless, the interchange rates are easily googled and differ somewhat based on merchant industry and size (market power again). Convenience stores successfully banded together and negotiated a lower rate for their industry. National-scale grocery chains also pay less. Again, not an example of a competitive market. The most effective means to cap rates has been legislation (e.g. the Durbin amendment and efforts in the EU.) because of the cartel-like behavior of the market participants to keep prices high and compete on "features".
[1] https://www.bloomberg.com/news/articles/2018-09-18/visa-mast...