as I understand, a wallet is not really a wallet, but a key. This suggests it can be shared between computers. Meaning if I have my wallet.dat copied from box A to box B and both boxes run the bitcoin app, box A should pick up the transactions from B and vice versa so both always show the same balance (not counting network latency). Is this correct?
If I understand BTC correctly that would be correct. It is essentially running your backup on another computer. This would go fine and dandy until you started generating new BTC addresses to receive with. Both wallets would start generating different addresses so coins received in wallet A would not show up in wallet B unless you made all the addresses in wallet A then mirrored it to wallet B.
Long story short, it's the same until you start generating addresses. This is why it is also important to back up your file periodically. If you generate a bunch of addresses, start receiving BTC from them, lose the wallet.dat, restore an old backup missing the new addresses, those coins sent to the new generated addresses that are not on the backup are lost to the void, never to be seen again.