Good day sir
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Both are not in contradiction, that's what I'm telling you. When you go to an exchange, then you are giving out your ID, which is normal. You have a trading interaction, your trading partner knows in most cases who you are. You also know that you were trading with an exchange, do you ? So you know about them, and they know about you, and both of you know that you and them were trading coins for fiat. What has this voluntary act of giving away ID information to do with the fact that the MONETARY ASSET is not going to propagate your information ?
Please feel free to head on over to Poloniex and hand them your Picture ID
..then over to AlphaBay to buy crack & guns.
That's a perfectly good illustration. Go to Poloniex and buy monero and withdraw them. Next, do a few transactions to yourself with different addresses on the monero chain. (this is because monero is still leaking SOME information, it isn't perfect like zero knowledge proofs can be). Now, wait for a year or so.
Then, use those monero to go over to Alphabay to buy crack and guns. The link with your ID on poloniex is totally lost. Of course, one can try to trace you using opsec failures, IP addresses and so on *when you were buying on alphabay*, but whereever these coins came from (via poloniex) has become essentially untraceable through the monero chain, and that's all monero is concerned with: not propagating earlier transaction information.
Now, do the same exercise with bitcoin. See the difference ?
Why did I say that you had to wait for a year and that you had to transact a few times yourself ? This is because the monero anonymity mechanism isn't total. Poloniex knows what transaction it used to send you the monero you withdrew. If immediately afterwards, you do a next transaction to alphabay, then this transaction will only be mixed with a few other potential transactions, and if this transaction is found out because the FBI got into alphabay's computers and found all their transactions, they know this transaction too, so the "number of potential guilty persons" is limited (the anonymity set is restricted). However, the longer you wait, the more your Poloniex transaction is going to mix up with other transactions you have nothing to do with. If you have done a few intermediate transactions, you raise the anonymity set size essentially to the power of the number of intermediate transactions. After a year or so, this web of potential people is so large that this information is essentially useless for law enforcement.
However, what remains of course is all contact you take with alphabay, the delivery of the goods and so on. All this has nothing to do with the monetary asset used to pay for the stuff. It is not because you use a monetary asset that doesn't propagate your ID, that you don't leak your ID in another way. If you put your postal address to Alpha bay so that they send you the package to your home, of course that has nothing to do with monero. You'll get caught just as well. It is very difficult to do anonymous business. That's not the point. The point is that the payment system doesn't propagate information. Monero, in the example you asked for, doesn't do so. Bitcoin does. Unless you use bitcoin tumblers (you trust), and you do a lot of bitcoin transactions in between.
But if you do the same thing with bitcoin, that is, you buy bitcoin on coinbase, you do 3 transactions to yourself, you wait for a year, and then you buy a gun on Alphabay, and alphabay gets compromised by the FBI, then the link between your coinbase ID and the transaction is in fact very obvious. Bitcoin propagates it like a bush fire.