How is the monetary aspect of a bitcoin stored?
There is no such thing as a Bitcoin in normal Bitcoin usage. Bitcoin is a currency but it does not have the equivalent of a physical coin that passes around. Bitcoin is really a unit of measure not a coin or anything much like a coin.
Where the monetary aspect is stored depends on where you look.
Bitcoin full nodes such as the Bitcoin-core wallet, send and receive data to other nodes that are primarily blocks of the blockchain. Every full-node maintains its own copy of the blockchain. As you know, the blockchain is simply a list of transactions and the information about a transaction is only a list of input amounts and a list of output amounts. Each amount being associated with a script that usually contains a Bitcoin-address which is almost always under the control of some unidentified person.
As you know, to work out the total unspent amount of money currently associated with an address you can look through the whole blockchain, add an amount everytime that address appears in an output and subtract an amount every time that address appears in an input.
Since people nowadays use a new address for every transaction, to calculate a total for a person you'd have to know a list of addresses under the control of that person. That information is not in the blockchain data but is stored in the person's wallet (and not shared with anyone else).
A wallet could choose to have a file that contains the person's name, or an account-number and the number representing the total amount of money controlled by that person or account. But whether or how wallets do this and how they store this data is not really part of any Bitcoin definition or specification. Each wallet could choose to do this or not and to do it in a unique way.