What is referred as 'coin'?
A lot of Bitcoin terms don't really make a lot of sense if you look at them closely. There really aren't anything truly like coins.
As Mike answered, the nearest thing to a coin is a transaction output, specifically an unspent transaction output (UTXO). But if this is a coin, it is a type of coin that only gets used once and then is circulated no further but melted down and cast into new coins. It is a type of coin where almost all coins have a different face value. This is a very weird kind of coin.
As I understand it, a coin mixer works by taking as inputs a large number of unrelated UTXOs from a variety of unrelated people and, to follow the coin analogy, melting down those coins and re-casting the total amount of metal into a variety of new coins that are paid to a large number of new addresses. An outsider can't tell which input coins relate to which output coins because the transaction data doesn't keep track of that information.
So coins don't exist in anything like the form of a normal coin that passes unaltered through hundreds of transactions, which is fungible (indistinguishable from other coins), which is contained in only one wallet or pocket or purse at a time.
If a UTXO is a coin, then every coin exists in everyone's wallet at the same time. Every full node contains every UTXO.
The fungibility of Bitcoin does not come from the fungibility of the coins - every UTXO is distinguishable from every other UTXO. It comes from the melting and recasting process.
In a similar way:
- Bitcoin wallets are not like wallets - they don't really "contain" money.
- Bitcoin's "ledger" (the blockchain) is not like an accounting ledger - it is more like an accounting journal.
- Bitcoin balances are not calculated by the network, not by miners, not by other people's wallets - they doesn't need to.
- Bitcoin addresses don't identify locations or people. Things are not really sent to addresses.
- Bitcoin keys don't really open locks.
All the things I listed are really just analogies. Like most analogies, if you look at them too hard they don't make sense. So don't look too hard, be happy.
Why do people write about coins? Because it is a convenient shorthand, more familiar seeming than UTXOs - We use the word coins even though coins don't really exist in any especially coin-like way.