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Basically the claimant is asserting that fat people are fat not because of genetics, but because they overeat and therefore have no self-control.

To emphasize their point they reference concentration camps and the apparent dearth of overweight prisoners who are purposefully being starved to death by their oppressors.

The implication is that if fat people took the extreme position of starving themselves (or exercise some discipline) they would not be fat. Of course, this is medically dangerous. They may weigh less, but they could also die in the process. Hence the claim is false and deceptively simple while medically unsound.

Thus, the claimant is guilty of a Weak Analogy Fallacy.

From Dr. Bo's Logically Fallacious:

Weak Analogy
(also known as: bad analogy, false analogy, faulty analogy, questionable analogy, argument from spurious similarity, false metaphor)

Description: When an analogy is used to prove or disprove an argument, but the analogy is too dissimilar to be effective, that is, it is unlike the argument more than it is like the argument.

Logical Form:

X is like Y.
Y has property P.
Therefore, X has property P.
(but X really is not too much like Y)
Example #1:

Not believing in the literal resurrection of Jesus because the Bible has errors and contradictions, is like denying that the Titanic sank because eye-witnesses did not agree if the ship broke in half before or after it sank.
Explanation: This is an actual analogy used by, I am sorry to say, one of my favorite Christian debaters (one who usually seems to value reason and logic). There are several problems with this analogy, including:

The Titanic sank in recent history
We know for a fact that the testimonies we have are of eye-witnesses
We have physical evidence of the sunken Titanic

To me, this can simply be categorized as someone's ignorance of how genetics and the environment interact. Anyone on calorie restrictive diets are not exceptions to the rule; they are subject to the same genetic pressures as obese people with similar genetic instructions. The difference is that the environmental pressure of excess calories is not triggering the genetics that result in slower metabolism, the irresistible desire to eat more, how one's body stores fat, etc. So in the commenter's attempt at a witty, pseudo-intellectual retort, they are just displaying their ignorance.

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This comment is fascinating.

"Ever see an over-weight or obese person in POW or concentration camp photos and videos?
Apparently, genetics and other fat-factors don't work there."

It's obviously ignorance about basic human biology and genetics. It ignores the fact that genes in this case cause a predisposition, and is not deterministic. Having genes that predispose someone to becoming obese still needs food intake with a lot of calories. This is called a gene(s) - environment interaction.

But because this is not the place for lectures on human biology and genetics I won't go any deeper in the details about that. This is about fallacious reasoning.

The argument given seems to argue that obesity is not genetic because fat people are not seen in concentration camps or in a POW. However, because of the lack of any other context of the discussion I can't do justice to the argument though.

Well for fun let me reconstruct the argument in a basic syllogistic form, which in this is in modes tollens.
P1. IF Genes (and other fat-factors?) cause obesity THEN People with these genes are always obese.
P2. On pictures of a concentration camp or in a POW people are never seen that are Obese or over-weight.
C. Therefore, Genetics (and other fat factors) don't cause obesity.

This translates to;
P1. IF A THEN B.
P2. NOT B.
C. THEREFORE, NOT A

In this case premise 2 is false and thus the conclusion does not follow from the premises. Which is a non sequitur.

But because I have not seen the discussion I am likely making a straw man of this argument.

But without writing it in syllogism the argument is still fallacious.

The analogy with people in a POW or concentration camp is a very extreme case and is therefore a false analogy because anyone, no matter their genes, can be starved. Also using the POW and concentration camp as an example is very emotionally loaded, so I would also call this an appeal to emotion.

And lastly because I get the sense that this person does not understand how genes could in one case influence obesity and in the other case still can the same case the person could still be starved, this could also be an argument from incredulity.

false analogy
https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/181/Weak-Analogy
appeal to emotion
https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/29/Appeal-to-Emotion
argument from incredulity
https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/196/Argument-from-Incredulity