A hundred years ago, perhaps, you could find similar services for snail mail. You mail the letter to the "agent." The agent opens the envelope and finds the message you want forwarded, along with an envelope addressed to the recipient or a separate slip of paper with the adress to which you want it sent, or just a sealed envelope addressed to somewhere with no return address. The agent puts the message in the separate envelope that you sent if necessary, or creates a new envelope using the address you included if that's the chosen method, and drops whichever "final" envelope results into a mailbox. The postmark shows the letter coming from where the re-mailing was done, and if the agent destroys any other bits and pieces he received, there is no way to trace it back to the original sender - except by getting the re-mailer/agent to reveal the information.
(Much US mail isn't "postmarked" in recent years, it's just "cancelled" and doesn't show an origination anyway.)
The awareness of this system is what makes some of the Nigerian Scam letters remotely plausible if you happen to be a total idiot.
There are various laws, treaties, and regulations that require anyone operating a business web site, and especially those doing stuff of this ilk, to maintain logs of message transactions, so in cases where the rules are observed it remains possible for legal authorities to demand the logs and to identify the original sender.
There are, unfortunately, a few places where the rules are not observed (i.e. where criminal governments1 exist) and a forwarding service sited in such a place could, in fact, provide a "secure anonymity."
1 A personal opinion with which some "diplomats" would argue.
My understanding is that Borked has been forced to reveal logs in at least a few cases, but even when the logs are obtained it may be necessary to reconstruct an association between disparate bits of log data from multiple logs to extract the needed information, so it may take some bit of "interest," and some skills, on the part of the authorities to actually determine the identity of the sender of a specific message. The amount of "analysis" needed is often fairly trivial, so it takes only a little bit of "interest;" but the difficulty can be substantial under some conditions exploited by professional criminals.