I wasted years of my life in crypto

But you (theoretically) cannot know who mined the coin, or who is actually the holder of the coin, thus the anonymity. Though currently this is getting restricted as governments require more ID verification from businesses dealing with crypto, which links up your coin to a real person.

The correct term here is pseudonimity - you know the immutable, stable wallet id of who mined the coin, which is a pseudonym for a real person. Anonymous systems are ones in which it's impossible to associate an identity with the work item.

For example, if I send cash through the post office, and I don't sign the envelope, that is a form of anonymous payment - it's impossible to tell who sent the payment (assuming there is no footage of the post box where I deposited the envelope, and I left no DNA on it, etc). If you receive a second payment, it's impossible to tell whether it came from the same person or someone else.

How is the serial number on cash any more anonymous than bitcoin addresses?

Because it's attached to the bill itself, not the owner or the wallet.

If I give you a dollar bill with the serial number 100100, it's impossible for you to prove that bill came from me (unless you have forensic evidence of me giving it to you, of course - but that's equivalent to having photo evidence of me typing in my private key to a BTC wallet) . If you find a dollar bill on the street, it's now yours, you can't know anything about its previous owner.

In contrast, a BTC address is a unique identifier for someone who owns the BTC. The blockchain stores all addresses that it ever interacted with, so even if you create thousands of wallets, they can all be-anonymized quite easily if one is, as you can track how money was sent between them.

You don't have to transfer bitcoin. You can give someone private keys to the wallet and they can do anything they want with it. It would be exact equivalent of giving someone the bill

You do have to transact bitcoin to get bitcoin into the wallet. Plus, you can't prove to someone you haven't kept a copy of the private key, so you can't really transfer ownership of a private key, not trustlessly.

Let's assume we know a certain btc address belongs to Alice and the other one to Bob. If Alice transfers coins to Bob's address we can see that Alice transferred ownership of the coins to Bob.

But if Alice just gives private key to her address to Bob, then Bob generates new address (which we won't know is his) and transfers the coins there when we won't know for sure that the ownership of the coins changed. If we didn't see Alice passing the private key to Bob we have absolutely no reason to think that Bob owns any coins. We see that his known public address is still empty.

At some point, Bob will want to spend the coins on something that he needs. At that point, you'll be able to trace the whole chain of transactions and know that Bob got the coins from Alice. Sure, you won't know that Alice transferred the private keys to Bob, but you'll still see a chain of transactions that starts with money in a wallet associated with Alice and ends in a wallet associated with Bob. The private key transfer doesn't achieve anything at all: Bob could just as easily have opened a new wallet and asked Alice to transfer money there instead of his known wallet, nothing in the analysis would have changed.

True, but only if you monitor Bobs purchases. Funds are anonymous until you see them leave network. And that might be years or decades in the future. And one sale/purchae on uncontrolled exchange breaks the chain.

The chain is unbroken. If at any point you identify the owner of a wallet, you then find out the full transaction history of that person. That is the problem with putting all of the data in an append-only ledger that is pseudonymous.

Yeah, try to tracm down chain that went through russian crypto exchange wallet. Then Thailand then Venezuela.