Keep in mind I haven’t seen any episodes of season 8 yet
Archer's father, at least biologically, is irrelevant. Archer is, in fact, a laboratory creation made by the American Government in a Super Soldier Spy program
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Despite his foolish, and sometimes animalistic behavior, it has been said time and time again that Archer is/was a fantastic field agent. More often than not he accomplishes his task and is quickly able to adapt to any situation. His looks are also, though perceived as very attractive, are relatively plain; plain skin, plain hair, and no distinguishable features, meaning he could blend in many places and it’s shown that he could pass for olive-skinned ethnicities such as a Latin American or Mediterranean, in addition to many of the primarily paler caucasian countries.
Speaking of looks; Archer has the body, endurance and survivability like no other in the show. He particularly adores his good looks and great features but his "perfection" is why he is able to survive such feats such as being shot over 30 times over the course of the series without dying or receiving permanent damage other than the occasional scar. He's also able to metabolize toxins better than anyone, in case an enemy tried to poison him; which is why he can drink more than anyone else in the show (except maybe Pam). Not to mention he is capable of some incredible things like scale an entire building in a matter of seconds (ep. The Honeymooners) or be revived after drowning without irreparable brain damage. Also, his immune system is top notch which is why he is able to contract a variety of diseases (STD, malaria or otherwise) and be easily cured or have little effect. The cancer he received is obviously from, as we have seen, incidences with exposure to high radiation that would have certainly killed a normal human being. He may be super, but he does have a limit. Though as a super soldier, Archer is (mostly) incapable of fathoming his own mortality. With the exception of the cancer, which would have caused a relatively slow death, any other brushes Archer has had with death is seemed as rather careless, as would be desired for a spy.
Additionally, his intellect is top notch. It’s been shown that Archer has learned 9 languages: Russian, Spanish, French, Romansch, Italian, Arabic, Yiddish, Portuguese, and Finnish (and one phrase in Urdu), so he’s able to pick up on languages fairly well. Furthermore, it’s why he can make very quick mathematical calculations off the top of his head, and count bullets being fired. He is also the largest literary scholar out of all of the characters because he makes more references to literature, culture, and history than anyone else on the show.
Aside from his anatomy, it lines up with Archer's past. Archer's past is often unknown to cloudy at best and there are very little details about it; that's because most, if not all, of his memories are either implanted there or just how he remembers it. This is why he doesn't remember being "dressed as Hitler" for Halloween, or that he didn't see his mother for basically the first few years of his birth, or even that his "birth" was in Tangiers-- which probably means he doesn't have a birth certificate, he was never really born. Additionally, that's why he perceived his childhood as being spent at multiple boarding schools where he was often alone; it's because he didn't go to boarding school, he grew up in a lab with no one else around his age. If there were any incidents of Sterling being around other kids, it seems like they were with bullies, which could have been a test to see if his specialized training was in effect yet, every time there was a “failure”, they would “bring him to a new school” every few years to retest the super soldier spy’s capabilities.
So who created Archer? Well, thinking about anyone else in the show who is able to build life from little to nothing the obvious answer is that Archer’s creator, or “father” if you will, is none other than Dr. Krieger… Sr.
Being a Nazi scientist, it could've been that Krieger Sr. did some tests on human genetics and anatomy in the Holocaust (yes, dark I know) and was one of the scientists drafted in “Operation Paperclip” and was contracted by the CIA in 1947 to create such a “perfect specimen” for the incoming Cold War. However, like many operations of the American Government, it was eventually shut down, like MKUltra or Acoustic Kitty. Not being able to create human life and possibly in danger of being somehow thrown under the bus by the CIA for Crimes Against Humanity or otherwise, Krieger Sr. fled to Brazil, like many other Nazi’s after WWII wherein he more so “created” a family, including Algernop Krieger, until he was discovered and killed (indirectly) by Mallory Archer.
Now, here is where Mallory fits into the picture. It’s obvious that Mallory has worked in espionage all of her life and she is extremely intelligent, so it wouldn’t be hard to believe that she was involved with the Super Soldier Spy program, probably as some sort of trainer, caretaker, or any other sort of person that could constitute as a “Mother”. Now, the story of Archer’s birth in Tangiers is either completely made up as a cover story, or, though maybe less likely, Mallory really did have a child but it didn’t last long after childbirth. So being the mentor and overseer of this young “perfect spy in the making”, Mallory grew matronly over Sterling until he was nearly 18 when the operation was shut down.
The cause of shutting down the operation may actually be due to the “crazy weekend” Mallory has referenced where she allegedly slept with Nikolai Jakov of the KGB and Len Trexler of ODIN (and Buddy Rich), however, being in a blitzed out state, it could have been that Mallory actually blabbed about the operation to them. By leaking information, the operation would have been shut down and likely the test subjects would have been terminated. In the episode Once Bitten Sterling remembers the time he was about to go to Johns Hopkins on an athletic scholarship but is then gunned down in an assassination attempt by a “crazed fan” wherein he loses his scholarship. This was actually the U.S. Government attempting to put the subject down, however, because he was able to withstand the gunshot, Archer survived and was rehabilitated by Mallory. Having a mother’s love and feeling such a guilt about nearly having Sterling killed and shutting down the project, Mallory took Sterling in and gave him the phony identity as her son and left the CIA wherein she created ISIS, she may be a bit of a bitch, but she isn't heartless. For the next 40+ years she kept romancing Len and Nikolai in hopes of keeping their mouths shut about accidentally leaking the information. Additionally, she also went to Brazil to kill Krieger Sr so that there were no others that new about Archer’s true identity.
However, Mallory is a stubborn and greedy person, so this is when she kidnaps Algernop in hopes of having a scientist as brilliant as his father and potentially create more super spies; where we have seen his failed clones in his lab, but due to advancing technology, he moved to bionics with the creation of cyborg Katya, Conway’s arm, and Ray’s lower body and hand, all in hopes of making a powerful intelligence agency. This is also why she had Lana use Archer’s sperm to create Abbiejean, to Potentially breed another “perfect specimen” into a future spy. Additionally, Mallory kept pestering Archer to get a vasectomy because she didn’t want to risk countries around the world having Archer’s “Super Genes”.
Also, why would a lab creation have a belly button? Obviously a feeding/nutrient tube while in test-tube utero; simple as that.
And Brett Bunsen was an earlier test subject, which is why he could survive many gunshot wounds without dying, but was severely affected until he was shot in the head and finally died.
The vision in Once Bitten where Archer remembers seeing his father could easily be a dying hallucination or dream right before he gets revived with the anti-venom. The possible father mentioned in Lo Scandalo may have simply been Mallory testing Savio Mascalzoni to see if he would give himself away for knowing about the government operation and Archer’s true origins.
So perhaps, at best, Archer may have been from the sperm of a donor, but that barely constitutes as a father, or at least not what Archer is looking for.
TL;DR Archer is a government laboratory creation, created by Dr. Krieger's father.
(Also, MatPat, if you’re reading this, I love your shows, but please don’t take this theory from me)