How in the world would the codes April knows work on the new Enterprise twenty years later?
Presumably, the ideas are that (1) it wasn't just a backdoor to his Enterprise, it was actually in the core OS of all Federation starships and (2) it was never deleted, possibly because someone else who knew about it [Admiral Marcus? Section 31?] managed to prevent it being changed.
Hell, it may not have been April's *own* backdoor, but something he discovered that was already there.
Like say some sort of 5 digit computer code which can allow a person to hack another starship and oh I don't know make it lower its shields