I hear people talking about anonymity with various currencies, eg Monero, or DASH, but as most people seem to use exchanges, there really isn't any anonymity, because the exchange presumably knows each person's identity and therefore has a record of all transactions, whether they be for bitcoin, or any other "anonymous" currency. Is this right? It would seem that anonymity is a function of the exchange and not the currency (granted there are differences in currencies that impact anonymity as well).
If the above is true, are there any legitimate (I know this is subjective) exchanges that do not keep identities of their customers? I have heard of local meetups, but that does not seem very convenient and there are trust issues.
not necessarily.
there are a lot of exchanges that don't require any information on their users. most altcoins exchanges are like this. you deposit bitcoin and buy any altcoin you want and apart from your IP (which can be hidden) and the bitcoin address you deposit from nothing else is going to be revealed.
and also it is the feature of those coins that allow more anonymity. I am not an expert in monero and those others but you can read their paper and ask around in their topic for more details about how they do it. but know that it is the coin itself.