The UAC in Doom is more tongue-in-cheek compared to many past versions of the Union Aerospace Corporation (with exception of perhaps the Doom RPG series and some of Doom 3). That is to say it is filled with a lot of self-referential humor about the Doom series in general (Doom 3 was more ironic, for example the poster at the beginning suggesting that UAC was unlocking the secrets of "Heaven", while in fact they were actually researching Hell).
If the 2016 reboot exists in the same universe as the rest of the games, by the time Doom 2016 takes place UAC isn't even bothering to 'cover-up' the fact that they are experimenting on hellish creatures, and artifacts and travelling to Hell. In fact the new business model is clearly inspired on a occultist Satanic religion, somewhat like a secret society (of both science and mysticism). Whereas earlier games and in particular Doom 3, no one except maybe the top researchers knew what was going in the advanced research, everyone seems to know, even possibly people on Earth by this point (or at least its brushed away as rumors and is the source of popular entertainment 'games' of the day, and in the past). That's not to say that some details are still classified, but no one seems to be shocked about existence of Hell in the game. Whereas only a few years before (if Doom 3 is part of Doom 2016's universe), they were still covering up all events of incursions from Hell, and the actual nature of their research.
Due to this self-knowing nature of everyone and everything in this universe its hard to say what are clear easter eggs in some instances, or actual artifacts to make this self-referential world more consistent with itself.
Many elements of the base seem like highly stylized versions of the Doom 3 style architecture, logos and companies included, but further updated even more technologically advanced. You can find other elements that seem to link to Doom 3 including the basic date of when artifacts were found on Mars and later in Hell (Doom 3 used the date of 2104 for finding in a cavern on mars, one of which allowed them to unlock secrets of portals to hell, and the 2016 reboot maintains that date), as well as roughly sharing the 2095 date for start of some of the main research on Mars (Alpha Labs and Argent Energy respectively).
Finally there are other minor references such as copies of Super Turkey Puncher 3 existing in both Doom 3 and Doom. Like in Doom 3 and in Doom RPG, "Doom" (original series exists as a kind of known video game series to the people of the future, that mysteriously mirrors the 'accidents/incidents'), likewise a classic Doom movie poster can be found in a locker room in the entrance to the Administration building of the Advanced Research Labs.
Finally many of the details surrounding the mysterious main character the DOOM Marine seem to be self referential of the main character of most of the Doom series, but of someone who decided to remain in Hell (similar to the ending of Doom 64, and there may be other Doom 64 references in the game as well, including the references to the mother demon from that game). There are other references to original Doom and Doom II including the Icon of Sin.
Like the previous installment, the UAC is wholly aware of Hell but was conducted to continue in researching and producing the Argent Tower and exploiting Hell's resources as it was in "Humanity's best interest to do so" as quoted by Hayden himself.
When Olivia unleashed a wave of Lazarus energy that consumed and killed many of Hayden's staff and releasing apprehended monsters into the labs, Hayden was forced to release the Doom Slayer from restrained slumber as he was the only one that would stand between Hell and Earth. Despite the Doom Marine's distrust of the UAC and the destruction of his Argent Tower, Samuel Hayden accepted the Marine's beliefs and continued regardless to assist him, including giving up the BFG-9000 prototype that was being developed.
There are a number of UAC related Logos in the 2016 reboot, including versions of classic Doom UAC logo (sometimes with Union Aerospace Corp written underneath, and sometimes without) and variations based off the Doom 3 UAC logo. The new design of the logo crest actually looks like its highly inspired by the Doom (film) logo crest, as well as the use of U Logo C design looks highly inspired by the film logo..
The new UAC logo appears to take elements from the classic logo (which can still be seen on various UAC products throughout the base) The discs have become stylized U and the C, and the Doom 3 triangular shape becomes a stylized "A" in the center where the triangle is in the classic logo.
There is yet another logo that appears on space related vehicles which basically a ringed planet with an orbiting satellite or moon, with the older style Doom 1 triangle/arrow in the center of the planet.
There is at least one Logo where there is a highly stylized word "Union" listed bold block letters. Some versions where Union Aerospace (without the Corporation attached).
Beyond that there are many other logos with their own unique designs.
It may suggest that the different logos refer to different divisions and sub-corporations of UAC.