what does "mixed with other coins" mean exactly? If I own a wallet, and someone steals coins and "mix" their coins with mine in my wallet, they eventually have to get those coins back right? So I probably know them and can found and interrogated?

legit question

The problem is that 'coin' is not a useful analogy and it makes things hard to understand. Coins don't exist. Only balances do. What you have is a balance (the number of 'coins'), but coins don't have an identity (as they don't exist). What one can track is the transactions.

So if you send (part of) your balance to another account, that can be seen. If that account sends on a similar amount, that also can be tracked. (Even if you split it between multiple accounts by sending to multiples of them.) The only way you can lose sight of the money if goes to an account that has a (relatively) large number of incoming and outgoing transactions. That can be a person actively paying and earning in BTC (and then you caught whoever stole the money) or a mix. It collects incoming transactions from a large number of addressess (accounts) and then sends these out to a large (but differing) number of other addresses. All you see is constant incoming and outgoing transactions of different values, that add up to (almost) 0 over a long period of time. But you won't be able to correlate them (and no coins exist that could have an identity, just balances).

EDIT: Side note - the way actually BTC works, they don't store your balance (as, I think, ETH does), i.e. the balance of an address (account), only the transactions. The balance can be calculated from the sum of all the transactions for that specific address (incoming ones add, outgoing ones subtract from it).

You can track the money going from the cold storage wallet into an account that the mixer service owns. Say it's 10.000 coins.

The mixer then transfers 37, 185, 205, 1002, and other random amounts to other accounts, which in turn transfers it to other accounts, and at some point they get funneled to one or more accounts owned by the person who originally transferred the money into the mixer.

Couple this with a lot of other people doing the same at the same time for the same mixer service, and you cannot say who owns the coins being transferred between the accounts. It is public what money was transferred back and forth, but without some serious analysis it's practically impossible to track who is likely to own the accounts where the money end up.

Let's say there are three BTC addresses in the world, #1 to #3. Let's say address #1 has 1 BTC that you're interested in.

On Monday, all three addresses send all their BTC to address #4. The next day, address #4 sends the coins back to addresses #5 through #7.

Which address out of #5, #6 or #7 has the original BTC you were interested in?

So - and thank you for bearing with me on this - individual BTC don't have a unique identifier that allow one to trace its transaction history? I must admit, I always assumed that they were uniquely identifiable.

Yes. In the first place, BTC are divisible to eight decimal places, so there would be a lot of these IDs if they were to exist. And there's no reason the divisibility can't change in the future, so these IDs would have to be quite complicated.

A (non-coinbase) transaction is valid if its inputs equal its outputs, and if its inputs come from the outputs of other transactions, which in turn are valid if their inputs come from other transactions, all the way to one or more coinbase transactions. This is sufficient to demonstrate that the transaction is valid - ie it's using coins that exist instead of creating them from thin air. That is all that the protocol cares about.

Thank you for taking the time to clear up my misapprehension.