Can I “duplicate” a wallet using a backup phrase (12 words)?
Yes
Where is my balance stored?
Both wallets will scan the relevant parts of the public transaction-journal ("blockchain") to add up the amount of money controlled by the keys derived from your backup phrase - this total is your balance.
Wallets are very likely to store this total locally, as a convenience, so they don't have to re-scan the blockchain and add up the total all over again every time the wallet is opened. Exactly where and how this number is stored is likely to vary depending on the software used. The storage of this total is of no real consequence.
if I send from (A) to a friend's wallet (Z), would my reduced balance also show on (B)?
Yes, both wallets will learn about new confirmed transactions that affect your balance.
Wallet A might show the transaction amount as pending until it receives a confirmation, a confirmation is just a new block appearing in the public transaction journal that either contains the transaction in question or which builds on top of such a block. Wallet B learns about the confirmation in the same way and will similarly notice that the block contains a transaction that affects an amount associated with an address derived from your backup phrase. The mechanism is the same for both wallets, Wallet A has no special privileges.
To elaborate a little:
As Murch noted in a comment, wallet A might have some additional information that doesnt get transmitted to wallet B. This is information which isn't transmitted to the rest of the Bitcoin network and which has no effect on validation of a transaction or on the mining process by which payments are considered to be confirmed. One example is if you choose to add a memo "payment for new trinket" to the transaction, this memo is not transmitted to anyone else and so is not known to Wallet B (or anyone else). It is also lost if you destroy wallet A and re-create it (e.g. by deleting an app and associated data and then reinstalling the app and using the old backup phrase). If your wallet A allows you to organise transactions into accounts, wallet B wouldn't automatically know about those accounts. Similarly, if wallet A is a Lightning-capable wallet, Wallet B wouldn't normally know about the lightning transactions that Wallet A made, only about the final on-chain settlements made e.g. when a lightning channel is closed.
However Wallet B should still normally show that same total as Wallet A for total amount of Bitcoins that the basic Bitcoin network and it's public replicated transaction-journal ("blockchain") consider to be controlled by private keys which are in your wallets A and B.
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