Batman beat him to it. I guess there's no reason that there can't be two superhero hangouts on the moon, though.
It does sort of enter the realm of how many places do these guys need, though?
I mean, Superman has the Fortress of Solitude in the Arctic, the new "SuperDad" Fortress in the Himalayan mountains, and of course the Justice League Satellite. Batman has the Batcave, the Justice League Satellite, and that base on the moon. I've got to think a lot of resources go into maintaining these things. It's admittedly easier for Superman because he's, well, Superman.
I also think from just the perspective of a "DC Universe", you wouldn't want to use up all the cool hangout ideas divided among just a few characters. I mean, though I didn't like the Lois and Clark comic much or the whole idea of SuperDad, and don't think the mountain Fortress is as cool as the arctic fortress, the mountain Fortress is cool in it's own right if it's not judged relative to the original. It would have been fine as a hangout for SuperDad in the "two Supermen" scenario where new52 Superman and Rebirth Superman co-exist, with the assumption that new52 Superman would get the Fortress he built, and was fine for SuperDad when they did co-exist and SuperDad was operating in secrecy. If the mountain Fortress falls, it opens up that location as a possibility for some other hero, after the dust settles and maybe in a different part of the Himalayas.
And so it makes sense for it to fall if the new52 Superman and the Rebirth Superman are going to merge into a Reborn Superman.
I am not a fan of the Chinese Super-Man, but if he doesn't have a Fortress of Solitude yet, he could have one in the mountains of Tibet (Well, maybe not, whether Tibet is a part of China or not is a controversial subject) or something since the mountain fortress motif would be available.
Or maybe the Arrow would hang out in a Himalayan Fortress as a secondary thing to the Arrowcave- Ollie is rich, he owns a helicopter, he could get it built in secret. In some continuities, he was trained by the Raj Al-Ghul, is an acquaintance or adversary of the Raj Al-Ghul, or actually even has briefly *been* the Raj Al-Ghul. The League of Assassins has a mountain top fortress. Maybe Ollie would like one, too. Where's the League's mountain fortress supposed to be located at, geographically?
There's also the possibility of having an Indian superhero who is actually from India hang out up there.
I just think allotting like half a dozen hangouts to each of your two top guys would be counterproductive. It doesn't even open up many extra storytelling possibilities. It's like, okay, Superman needs to get away from Metropolis or regroup from his enemies or have some time to think amid high Kryptonian technology and to keep things at, and so on and so forth. If the Fortress of Solitude doesn't serve for any given reason, or has fallen out of his hands as part of a storyline, he's got the Justice League satellite, and vice-versa (Granted, one is a shared hangout). If both fall, he could probably go to Bruce or someone and see if he can use their hangout. I don't know what having two fortresses really opens up storytelling wise- and since most Rebirth Superman's biggest fans are really post-Crisis fans (or prior) who want traditional elements, they are going to mostly prefer the arctic hangout, and New52 Superman fans are only going to have had the arctic version with their guy (Okay, our guy. New52 Superman is awesome), so that's the one to keep.
I think if Reborn Superman is who we think he is- the combination of New52 Superman and Rebirth Superman with a new back story with elements of all of Superman lore in it- it makes sense to sort of wrap up the elements that don't make sense in that narrative like the extra fortress Rebirth Superman needed back when he wasn't the world's Superman. You get to what this Reborn Superman would have had if he had existed all along, which is parts of both Supermen, and then expand out from there as he lives his own life going forward. I doubt we'd see many anachronisms like an extra fortress if they really want to just sort of do a one Superman who most people think has always been one Superman kind of deal. Because there is no reason for this hypothetical Reborn Superman to have an extra fortress in a revised timeline.
Of course, we don't know who Reborn Superman will really be yet, so all that's just speculation.