That makes a lot of sense. I'm not one to say that the showrunners are lying. What I think possibly happened is this: Loeb and Bell know the scripts and the ins and outs of all MCU TV shows. Because of this one or both of them could have known a while ago that the Agent Carter staff were gonna use the Secret Empire and that it would use the Arena Club as it's "front". The Arena Club of course uses the letter A as it's symbol which makes a lot of sense. The letter A does look, inverted, a hell of a lot like a Ram's Head. So here's my thinking on the matter: Either Loeb or Bell took that idea and ran it over to the AoS staff and told them what was to be going on in season 2 of AC and they noticed that it ties in well with the Zodiac (which I have been led to believe played a part in the comic book that AoS draws the most from, Secret Warriors). So ... they decided to use the Ram's Head (Ares) from the Zodiac as a feature in their show. Thus they, retroactively, are able to take something from Agent Carter and give it new meaning without forcing the Agent Carter team to be as restricted as they could be by later canon.
AoS is reactive to canon in the films and because of the nature of Agent Carter (where it is leading directly into the founding of SHIELD) it has to be the same with AC. The reason that AoS was able to do the whole Theta Protocol thing with AoU was because Loeb and Bell and the showrunners got access to the finalized script of AoU as soon as it was finalized so that the show didn't in any way contradict things that would happen in the film. I think it could be a similar case now. AoS showrunners knew what was going to happen in Agent Carter and because of that they were able to take something that wouldn't tie in well and add it into their own show. The fact that it is the Ram's Head symbol means that the Secret Empire doesn't necessarily have to be HYDRA or HYDRA-connected out right, but rather that the Arena Club and the Secret Empire could be Zodiac. We haven't gotten to Zodiac yet except in the Agent Carter One-Shot and that was only by name of a virus.
Basically what I think is that we may get Zodiac in the future and it is gonna end up being an Illuminati type organization. HYDRA, Leviathan, The Secret Empire, and others had members in the Zodiac and though they were all separate entities they had similar methodologies and goals. Thus, technically speaking, the showrunners could claim that the Arena Club isn't the same as the thing on AoS and yet it is (because it wasn't intended by AC to be, but AoS, retroactively, sort of "forced" it to be connected).
Sorry for the word-wall, but honestly I don't see how they (the bigwigs in the MCU) would let them get so coincidentally similar without it actually having some connection.