Hi Trevor, welcome to the community 
Monzo use a supplier called Wirecard who store our (users) funds in their accounts at the moment. You can see more details about that on Monzo’s T’s & C’s page.
But now that they have a full banking license, they will be able to hold our deposits themselves soon (in the Bank of England’s accounts, to be precise), when the current accounts launch.
Monzo aren’t owned by other banks. Their business has been funded by investments from venture capitalists, crowdfunding & independent investors -