It appears to be a Call of Cthulhu or BRP homebrew with influence from video game RPGs.
Based on my research, this character sheet is not for any published role-playing game, but appears to be a homebrew system. The biggest influence on this system is clearly a 1920s-setting Call of Cthulhu edition, but there are major features which most likely come from video games, not other tabletop RPGs. Specifically, it is most similar to Fallout 3 or Fallout: New Vegas.
My reasoning is as follows.
Call of Cthulhu/BRP hallmarks
A major influence on this setting is clearly Chaosium's Call of Cthulhu percentile-based tabletop RPG, one of the editions released in the 1990s or later, and using the default 1920s setting. Alternatively, it may be drawing on Chaosium's Basic Role-Playing, which is the rules system used for Call of Cthulhu.
The first reason is that the Intelligence attribute has a derived attribute called "Idea". This is the case in Cthulhu where the stats are generated by rolling 3d6, but since that would be too low to roll on in a percentile system, Cthulhu has an Idea stat of Intelligence × 5. "Idea" often appears specifically on Cthulhu character sheets, and on some sheets it is the only derived attribute so presented, so it makes sense why the player here would include it on theirs.
The skills are all from Call of Cthulhu. Spot Hidden is particular to Call of Cthulhu and Basic Role-Playing. Biology was added around the 1990s editions (previously there was Botany). Allowing for abbreviations (e.g. Library Use as "Library"), all of these skills are from that game.
The presence of a film camera and flash powder suggest that it is set in the 1920s, rather than modern day. Flash powder in particular dates it to that era, since it was invented in 1887 and began to be replaced with flash bulbs around 1929. Call of Cthluhu was usually particular with its historical accuracy.
The Professor class (or profession) and the sword cane are also particular hallmarks of Call of Cthulhu.
Attributes
Dr Trotzky's character sheet diverges from Call of Cthulhu in its attributes: Strength, Dexterity, Perception, Endurance, Faith, Charisma, and Intelligence.
Creative Reckoning's List of Attributes by Game lists no published RPG which has this attribute list. Only two use a Faith stat (Fantasy Wargaming, and World Tree, neither of which fit).
The closest match is the Fallout video game series, which uses the SPECIAL attributes (swap Luck with Faith, and Agility with Dexterity). Even so, that's not entirely conclusive, since you could also arrive at the same stat line by taking the six D&D stats and breaking Wisdom into Perception and Faith. If you look carefully at the sheet, you can even see that Endurance was originally called Constitution, which was erased and written over.
Perception also replaces an Accuracy stat. I once actually added an Accuracy stat to d20 for a campaign set in modern times, since Dexterity is too useful for both AC in a world without armor and ranged to-hit, hence I split it into Dexterity and Accuracy. It is also plausible that this game represents a split of Wisdom into Perception for spotting enemies and Faith for divine magic and saving throws (in fact, there's a Save stat next to Faith, which may or may not be relevant).
However, a much more significant resemblance to Fallout is that all attributes appear to have started at 5, had a number of points added as desired (5 in Fallout, 4 here perhaps by mistake or a -1 age penalty to Strength), and then points can be moved between them as desired. Maxing your Intelligence stat out at 9/10 is also a known Fallout 3 strategy because you can increase it by 1 later on to hit the cap of 10.
There is also a recent 2d20 based Fallout tabletop RPG which uses SPECIAL, but unlike the Fallout video games, it does not allow stats to be reduced below 4, and Dr. Trotzky has dropped his Strength to 2. Trotzky's character sheet is unconnected to the Fallout 2d20 rules in all ways except for attributes.
A second comparison to Fallout is that ability scores increase relevant skills by two points per point of ability. This isn't how the 2d20 Fallout tabletop RPG works (they have their own non-percentile skill system), but it is how the Fallout video games work; see for example this video for Fallout 1 and Fallout 2. Izmer Trotzky adds twice his Intelligence score to his Chemistry and Biology, and twice his Perception to Spot Hidden.
On the origin of the Faith attribute, while it rarely appears in tabletop RPGs, it is well known in the Dark Souls and Elden Ring video games, where it represents aptitude for divine magic. Those games also refer to Constitution as Endurance.
Other rules
A big point that people have noticed is that the numbers are way too low. Either this is not a percentile system, or it's a bad percentile system. Which is it?
The adaption of either Call of Cthulhu or Basic Role-Playing (essentially the same thing) is indisputable. It also varies significantly from those rules. That makes it either a homebrew or a third-party product. It's unlikely to be an unlicensed product, since Basic Role-Playing was not freely licensed until 2020, when the Spot Hidden skill was known as Spot.
That leaves either a homebrew system, or an officially licensed third-party RPG. The latter seems unlikely, given the changes to stat names on the character sheet, the haphazard hand-written character sheet, the inclusion of Fallout rules, and the heavy use of very specific Call of Cthulhu skills, profession, setting, and equipment, with the notable exception of Sanity. It also completely eschews certain Call of Cthulhu rules and makes the numbers way too low, which seem like rookie mistakes that you wouldn't expect from a publisher paying for an RPG license.
The most likely explanation, therefore, is that it's a homebrew system.
Throw is Str + Dex, Listen is Perception × 2, the derringer has 14% (to hit, presumably) which is consistent with Perception × 2, and the sword can also has "14", probably 14% to hit as a Dex weapon. Everything here is consistent with a percentile system where certain things are given a bonus of attribute × 2, as Fallout.
The low numbers may be explained that Fallout games usually start the player with such low skill numbers. One YouTube video on recommended character builds in Fallout: New Vegas shows a level 1 character with only 21 to 28 in most skills, and that's an optimal character build. A certain Fallout 3 character build video also shows a level 1 character with only 17% in Melee Weapons. For Izmer Trotzky to have only 24% in Spot Hidden is very poor by Call of Cthulhu standards, but it's a normal number for Fallout 3 or New Vegas.
Conclusion
In the past, I have played in RPGs where the GM was a big fan of some video game property and attempted to merge it with an existing RPG, hoping (and generally failing) to recapture the magic of the game. I'm reminded of this 2018 question where somebody wanted to adapt Pokémon to D&D 5e.
Given the evidence, it is most likely that this is someone's attempt at homebrew by cobbling together Fallout video game mechanics with Call of Cthulhu, resulting in a percentile game which would not play well because the numbers are too low.
Based on the system's likely inspirations, that someone is likely to be an American male in their thirties or so, and probably not the young airman who was recently arrested for allegedly leaking the classified documents. It was probably included at some point by complete accident as the documents were re-posted, and may not even belong to anyone who works for the US government. However, given that they probably re-posted classified material online, they are probably not in any hurry to come forward and identify their game any time soon.
