Are there alternatives for Europeans who dont want to be "purchase taxed" by VISA and MASTER CARD?

SEPA and Iban. It's as simple as it gets. Also much more secure considering it's a push payment instead of a pull one.
For easy payments, there is this: https://en.m.wikipedia.org/wiki/EPC_QR_code

A bank debit card strapped to the back of your phone. SEPA

N26, berlin-based, they handle the fees

Just out of curiosity, what purchase taxes are you talking about ? Aside from the yearly fee and the occasional (quite steep though) tax when I withdraw money too many times from non-VISA ATMs I don't think I have many extra fees with mine.
- Maybe you dont have too many extra fees, but when you use your visa card or MasterCard to buy from a merchant, 3% of the purchase price is paid by the merchant to visa/MasterCard. In the long run that puts prices up because the vendor is passing on those costs to you, the buyer. Its a tricky issue because if you pay cash, the shop has to store, transport securely, and pay to deposit the cash into a business bank account. That might cost around 3% of their takings too. The feeling here is that if you buy 100 pounds worth of shopping, it doesn't cost visa/MasterCard anything like 3 pounds to process that payment, more like a few pence. So perhaps we are being taken advantage of?
  
  I see, that makes sense. Y'all might be talking me into switching to Revolut. Though I've gotta say that Visa's insurance saved my ass this very weekend by handling everything to bring me and my gf home after a visit to the ER that left us stranded 800 km from home (even though we are both on the cheapest visa cards available in our country), so I'll have to see if they have similar stuff

If you're in France there's the Carte Bancaire.
Edit: I've just looked into Revolut and they're UK based with operations throughout Europe, so they might be an alternative for online payments across borders.
- Revolut is Lithuanian I think.
  
  They are based in the UK as stated by OP, but European clients go through a Lithuanian subsidiary

Yes,
After World War II and through recent times (well after the DM to € switch), Germany was one of the most cash-happy countries. If you didn't carry cash on you, there was a good chance you'd end up stuck and unable to pay for something. Even today, many businesses don't take credit cards compared to the US.
Using a card demands that you trust Visa, MC, etc. with your data as well as pay a fee.
The last time I was there (Christmas time) I was horrified to see the inroads these bloodsucking leeches made into Germany. People around the world are selling themselves and future generations into economic servitude and making themselves targets for corporate surveillance based advertising at an alarming rate.
Now would be a great time for a worldwide cash-based movement to spring up.
I ran a cash-based business for a decade and it wasn't so I could cheat on taxes (I paid those every year). It was so I could deny the CC companies their undeserved cut and remain congruent with my values.
Convenience or Freedom
Choose one.
- Some businesses in Germany only accept cards like giro and ec, so that's another possibility. In Portugal some businesses only accept MB cards. In Denmark some places only use Mobile Pay. It's definitely becoming more common for Mastercard and Visa to be accepted in all 3, but alternatives did exist.
  
  True, even before Visa and MC started making inroads, I was seeing EC cards commonly in Germany, but usually only for certain uses. The fact that some businesses now operate in a cash exclusionary way frightens me a little. We're headed for sci-fi dystopia.

Cash.

Interestingly, I just (10 minutes ago) learned about
https://en.m.wikipedia.org/wiki/Wero_(payment)
which seems to be in a fledgling stage but still further along than
https://en.wikipedia.org/wiki/GNU_Taler
- Danmark (and Finnland apparently, I think I've also seen it in Sweden?) also have
  https://en.m.wikipedia.org/wiki/MobilePay
  
  Here in Sweden most people use Swish

in Poland we have blik, and it's really popular
I make all my purchases online with blik, you can also use it in shops but it's a bit less convenient than paying with card
- I switched to Blik recently, and it's just slightly less convenient. Instead of just beeping a card, you need to enter the code from app and approve in the app. Takes 30sec more, but I'm happy for this to endorse Polish system.
- Blik has no chargeback equivalent

There is the CB network but working in France
- Breaker breaker 1-9; what’s my account balance?
  
  So dumb but I laughed o hard

Portugal has the Multibanco (MB).
https://en.m.wikipedia.org/wiki/Multibanco
It handles online payments, QR code and NFC payments in virtually every point of sales that support card payments, phone number money transference and operations in all the Portuguese banks ATMs.

Currently there are only alternatives online like Klarna (and iDeal in the Netherlands). The only way to do it in store is paying cash sadly
- Klarna has american connections. One of the biggest shareholders are "sequoia capital" 'merican
- What's so "sadly" about it? It's a solution to a problem, it's easy, and it's simple.
  
  Cash is insanely inconvenient, cannot be used at self checkout (which is the only reason I don't despise doing groceries), is more risk carrying big amounts and not even accepted above a certain amount in most stores.
  We just need an actual Visa/Mastercard EU edition.
  
  Because many times I leave home with just my phone.

Does twint have a mastercard like fee?