It's a complexity that probably won't be exceptionally visible in game events, but one that I think is useful to have in the background informing things, and it's very very fun for me to write. The statement that they are the empire is one that you could find lot of different perspectives on in the setting. Some could say they're the empire reborn, some that they're a empire but not the empire, and others could say they're a wannabe empire, or something else entirely with delusions of grandeur or really good propaganda. You could find people predicting that they're the seed of another true galactic empire, and others predicting that Empress Corrstei is a temporary figurehead who is going to have no successor.
Essentially, the confederation is different from the old imperials in that, while their rhetoric often suggests that the republic is illegitimate and should fall under the Empress' authority, in practical terms they do have real diplomatic relations with the republic and other nations, and participate in the galactic council. Imperialists don't allow any of that, and in the modern era of HW@ are essentially relegated to small pockets of piracy, having mostly either joined the confederacy or been killed by it. To this day they often refer to the republic as rebels, and the confederacy as rebels or usurpers.
Also, I'm loving seeing the progress lately on the assault frigate