Currently, Monzo is using prepaid cards (via Gaileo is my understanding) that are funded by a Sutton Bank account (as confirmed by my friends and the Sutton terms that US customers accepted at signup). I assume the reason for the prepaid card instead of a real debit card is partially due to the last second issues with contactless cards. Will we be issued a non-prepaid/real debit card once the card replacement flow is completed or will Monzo continue using prepaid cards for the foreseeable future? The reason it matters is because, being a prepaid card, there are certain classes of transaction that will not be possible such as renting a car, a hotel room, paying utility bills etc. I assume thats why Google was declining it when I was trying to pay for my Fi account earlier but could easily be wrong.
It’s worth digging into a few things here. Apologies if you already know a lot of this, but others reading might find it interesting too 
Galileo is our processor, they provide us with a connection to the Mastercard, Maestro and Visa Interlink networks which our cards use to make payments. They also instruct Sutton when to move money and how much in relation to these payments. Galileo also generate card details for us when we request them (e.g. the number after the BIN on the long card number, the expiry date etc.), and then pass these card details to our card manufacturers to make cards.
Sutton is our issuer, they hold the money in the accounts and move money between accounts when needed (either to different internal Sutton accounts, to Mastercard/Visa etc.) They are also the regulated entity in this case. Sutton own the BIN which our cards are issued under from Mastercard. This is the 6 digit Bank Identification Number at the start of your card number.
Monzo in this situation is the program manager - we interact with customers and manage the ent-to-end customer experience. Because we have direct relationships with all of these parties (Sutton, Galileo, card manufacturers, mastercard etc.) we are able to make sure the customer experience is how we want it to be. Some program managers take a very hands off approach and don’t have these direct relationships, however we do and very deliberately want the control to make sure you’re getting the monzo experience you deserve.
The cards we’re providing are debit cards, underpinned by a General Purpose Reloadable account at Sutton. This account has it’s own routing number and account number. It’s connected to the ACH network and is able to receive direct deposits. In the future this account will be connected to BillPay so you can pay your utility bills from the account.
That was a very long winded way of saying that as we build out our features you shouldn’t notice a difference between what our accounts offer and what you’re referring to here as a ‘real debit card’, and if you do and have examples please get in touch as we’d like to make sure this isn’t the case. The contactless part of this it unrelated, this is more just how the card interacts with the payment terminal in shops.
We’ve tried booking a hotel with the cards and they work fine. The only thing I could think of which might cause confusion at this early stage is if the hotel or card rental place didn’t like the fact the card doesn’t have your name on.
Hope that helps!
TL;DR Our cards should work in the same way as other bank cards and if they don’t let us know.