It was a tricky and sometimes mindbogglingly complex line for the Lincoln administration to walk. The last thing they wanted to do would have been to grant any shade of legitimacy to the government in Richmond, so they (as consistently as they could) attempted to deny the Confederacy's existence... referring (at most) to "the so-called Confederate states," and so forth. And yet, in reality, there was a functioning government, a military, a diplomatic service, etc. An outside observer not familiar with the details would have assumed the Confederacy was a separate government (as did Gladstone in a rather unguarded statement that he later came to regret).
One of the weird situations resulting was that the British government perforce continued to act as if the British consulates in the South reported through Lord Lyons in Washington, whereas in reality some of them went for years with no contact. Even when the Foreign Office wanted to send a rather stiff note to the Confederate government in early 1865, it did so through the U.S. State Department, which was unofficially asked to ensure the communication reached Richmond. (As the gist of the message was that the Confederacy should expect no help from Britain, Seward was only too happy to comply.) And the U.S. State Department continued to act as if Her Majesty's Washington legation was receiving information from its consulates, even though everyone knew it was a fiction.
(And it was not unheard-of for someone in the Lincoln administration to slip up in maintaining the fiction...)
Sometimes looks like "the Emperor's new clothes...." in order to deny the right of the group of states "styling themselves the Confederacy" to assemble into a government, the U.S. had to pretend that they hadn't.