Leaving to @Mithical's answer the matter of the pun, this is the ‘more boring answer’ about making of a seal with bread which makes the pun possible:
While the thing adhering to the letter to keep it closed is called a seal, it should be noted that the object used to impress the design on the wax can also be called a seal. Per the OED:
An engraved stamp of metal or other hard material used to make an impression upon wax, etc. affixed as a ‘seal’.
There is no reason to suppose that Dupin would not have sealing wax of his own at his disposal in his lodgings, or readily be able to obtain wax in the correct colour. What is wanting is the device representing the D── cipher with which to impress the wax.
As @PeterLeFanuLumsdaine points out, bread can be kneaded back to a more dough-like consistency, although it lacks the springiness of unbaked gluten. Depending on the nature of the bread one may not even need to add moisture to get a cohesive lump. Once such a lump has been obtained it could, feasibly, be sculpted into an imitation of the required sealing stamp.
In practical terms I can imagine that one might use the play-doh type substance over the top of one's own seal, or over a coin or button of appropriate size to give it the required rigidity for use, and perhaps coat the face with something to help it release cleanly from the wax. I wondered about lamp oil, but thought the scent might be too obvious, possibly black ink would work as an easing agent, and not be conspicuous on the black sealing wax required to emulate D──'s own seal.
Sealing wax is extremely soft at the moment one impresses it, setting quickly afterwards. The wax does not rely on the pressure of the stamping for its adhesion to the letter. A blob of sealing wax or a wafer[a] can be used between the layers of paper for the primary seal. The impressed wax seal then performs its primary function of identifying the missive as a genuine communication from the sender, although it also contributes to detection of tampering. A bread-play-doh stamp should easily be firmer than the wax, which is all that is needed to create the impression. It only needs to be robust enough to survive a single use.
To be clear, I’m not suggesting that this is something that was ever really done, or would work well enough to fool anyone. But I’m think it works just well enough to service the de pain/Dupin pun. If I’d had need to furnish a temporary stamp in my lodgings, I’d have carved it out of soap.
Edit: I present my proof of concept for the bread seal! Made with nothing but an M&S naan and a red tea light, with a little olive oil as releasing agent. Not perfect but the principle is proved I think.
[a]A small disk of flour mixed with gum and non-poisonous colouring matter, or of gelatine or the like similarly coloured, which when moistened is used for sealing letters, attaching papers, or receiving the impression of a seal.
