Theory: Charlotte and Code Geass are in the same universe, world and even timeline
(Spoilers for Code Geass and Charlotte)
There is the obvious fact that Lelouch and Yuu look very similar, but they also acted a bit similar, in that they were both kinda jerks who had no problems taking advantage of others. They also share similar poses, as shown below. (Sorry for the image formatting, best images I could find of roughly the same size)
So what does that mean and why does it have anything to do with them being in the same universes? Well, I have a theory that these two animes share the same universe/world.
In the last episode of Code Geass Suzaku killed Lelouch, thus he had (at least to our knowledge as of the year 2017) actually died after being killed via Suzaku, who became the new Zero. Given this, Yuu could be Lelouch reincarnated. It would also explain why in the first episode, Yuu seems to have similar powers by mind controlling some hapless truck driver for his own selfish desire. Later on, it is revealed that Yuu actually can copy others abilities by possessing people and use them as his own, which could be seen as an evolution in Lelouch's original powers.
Interestingly enough, the first power he has is controlling others much in the same way Lelouch did, but only for shorter periods of time. His eyes light up much in the same way that Lelouch's did.
It goes further than that, as Yuu also has a little sister and while she isn't necessarily blind as Nunally was, the fact that he has a weakness for his little sister could also be seen as a point in the favor of him being a reincarnated Lelouch.
I think the reason why there wouldn't be anything setting wise to link the two is a grand conspiracy. After the events of Geass, it is likely that Suzaku wanted to erase the knowledge of all the events that had partaken, which makes sense seeing how Suzaku was partly responsible for the bombings and killings himself that occurred. He could also be stricken from having to kill his best friend Lelouch and all the previous trauma from the war, so he could just try and live in denial like it never happened. Now the official explanation for why they were seizing people with powers in Charlotte was because they were using them for scientific research/exploitation, but this could all be part of suppression in case someone like Lelouch does ever come back.
(Very minor spoilers ahead for a couple Code Geass games, pertaining to the protagonists of that games powers just in case you wanted to go in blind)
The similarities and overlaps don't just end there, as there are a couple games called Code Geass: Portrait of Despair and Code Geass: Harmony of Ruin that has a protagonist whose name is: Charlotte Di Britannia. Just another interesting coincidence between the two? Well you would think so, EXCEPT this part from the wiki (which you can visit here if you'd like http://codegeassfanon.wikia.com/wiki/Charlotte_Di_Britannia):
[i]"Charlotte has the ability to bestow people with the power of Geass."[/i]
The comet Charlotte in the Charlotte anime did something very similar, so this may not just be an interesting happenstance. Charlotte Di also has other interesting powers, such as:
[i]"Her connection with Geass allows her to detect those people with the ability of Geass. The extent of this ability is unknown, but it does allow her to detect the location of her own contractors, even at a distance, and confirm who does not possess a Geass, regardless of whom granted it to them."[/i]
There have indeed been powers bestowed onto a character in Charlotte that can locate other people with powers, such as Kumagami, who could locate people with powers themselves. Given all of that, the similarities in the protagonists of both Code Geass and Charlotte, the amounts of potential coverups going on and other various correlations between the two lead me to believe that these two animes are in the same world, just at different points in time.