Online - Branchless banking is nothing new to the USA. Though the US may behind in other ways such as chip & pin, contactless, and ease of digital transactions.
Whilst Monzo is targeting trendy young urbanites in the UK, if they truly want to make an impact in the US, they need to go after a whole other market; The under and unbanked. This is a large market in the US for various reasons, This has some pretty large negative impacts on this market.
Due to the lack of modern digital transaction infrastructure, currently checks are still the main way many people are paid. Not just plumbers, electricians, maids, etc. without a steady employer electronic salary payments are not mandated either, and many even those with bank accounts choose to be paid by check. This means the only way the unbanked can access their cash are by utilising check chasing services. These are services which will cash checks for you at a hefty fee. Upwards of 10%.
There is no faster payments system either so third party tools such as Apple Pay, Cash App (Square), Venmo, Paypal, etc. are a must for instant payments. Everyone must be on the same platform to transact. Many banks, though not all charge fees to make digital ACH / EFT payments. These can take days to clear.
So if Monzo really wants to make an impact it will be to get the 9 million or so unbanked and 24 million or so underbanked persons in the US banked. Free checking accounts are nothing new, not all require minimum balances, so this is about making banking easy, accessible, and understandable.