The difference is that since cryptocurrencies' security is based on PoW, the more cryptocurrencies that are accepted by large numbers of merchants, the less secure each one is against attack.
Also, Mastercard and Visa are both spending the same currency. If you have multiple cryptocurrencies, the exchange rates would fluctuate with each other and it would be very inconvenient for the user.
There may be alternatives but just one is definitely preferable.
Mastercard uses EUR as a clearing currency and VISA uses USD, but the technology is more or less the same. Also, the debit and credit cards are not the money itself, they just link together the payment's initiator with the payment's recevier and they handle the communication between them.
The cards act like the mobile phone when you start a bitcoin transaction from a mobile wallet. They are the technical part of the system.
According to OP's example, it's like merchants will accept bitcoin, but they will use stickers on the shop's door to show what kind of mobile wallets (or bitcoin payment processors) they're accepting when it's about paying with bitcoin.