My understanding is that a wallet isn't a wallet at all*, it doesn't hold my coins, but its a key store; my coins are of course "stored" as records of the events on the blockchain.

Imagine I have bitcoins with Cloud Wallet Company (A), but I then setup Wallet Company (B)'s app. And let's assume they both use the same crypto algos to compute the seed from the 12-word recovery key, as per this answer.

https://bitcoin.stackexchange.com/questions/89834/is-the-recovery-phrase-generated-by-a-wallet-app-platform-independent

If I login to (B) and choose to import/recover my wallet and punch in the 12 words I got from (A), would I effectively now have two wallets? i.e. both (A) and (B) are setup with the same wallet.

Where is my balance stored? After entering my 12-words into (B) would their servers have to "run a query" on the blockchain to aggregate all the transactions and arrive at my balance??

And if I send from (A) to a friend's wallet (Z), would my reduced balance also show on (B)? Or would (B) cache my balance to avoid expensive query/compute from the blockchain, and so it would become out of sync?

Thanks!!

*Maybe I'm wrong but I think the wallet analogy only confuses noobs, since it sets up a mental model that's the opposite of the whole point of a blockchain.