Part 1

From the wording, albeit it is not entirely clear to me, I assume you mean discrimination of one group of prisoners by another group of prisoners and not singling-out one group of prisoners by their guards and the Nazis. The answer to both of those issues is, yes, that happened and, in fact, the two – discriminatory group practices by the Nazis within the camp system and inter-group discrimination – are strongly related and intentional.

Something that needs to be heavily emphasized at the beginning of this is that the Concentration Camps were a huge complex: The Nazis built over 42.000 camps in all of Europe with at least 1.100 of them being the big main Concentration Camps and their sub camps (this doesn't include the death camps e.g. or the Organisation Schmelt camps in Poland that are closely related to the Auschwitz camp complex but not actually under the administration of the Inspectorate of Concentration Camps). In January 1945, the SS counted a total number of inmates of all camps with 714.211 – and this is just the peak. Here is an unfortunately bad scan of a table from Nicholaus Wachsmann's book KL (my homescanner is shit for scanning books, I apologize) giving inmate numbers throughout the years of the existence of the camps. Some of these are counted continuously but the table shows that throughout the years from 1933 to 1945 literally millions of people were imprisoned in camps by the Nazis.

This is important in several respects for this question for a variety of reasons: First of all, these are people thrust into a situation that heavily relied on group discrimination and where positions of power meant a greater chance of survival (I'll detail a bit more momentarily); that came of many different backgrounds from Polish Catholics to French communists to German Jehovah's Witnesses; and who didn't suddenly stop having the convictions and opinions they had before. The Concentration Camps were a huge amalgamation of different groups and populations of Europe and, crucially, must not be seen as moral reformatories, in the sense that being or having been imprisoned in a Concentration Camps made these people victims of Nazi oppression but didn't turn them into shining beacons of behavior we'd consider moral. Italian and Austrian fascists imprisoned in Concentration Camps because of their opposition to German fascism at one point in time makes them victims of Nazi persecution but it doesn't change the fact that they were fascists and embraced a lot of the political ethos that entailed, for example.

The fact that it is quite possible to be a victim of Nazi oppression while still holding views and engaging in practices that are also oppressive neither justifies Nazi oppression nor does it mean that these people somehow "lose" their status as victims. A criminal imprisoned in a Concentration Camps without due process and being killed there in a brutal fashion is still a victim of Nazism and its systematic oppression because he was deprived of the process and civil rights – and murdered – in a fashion that completely goes against the ethos, rules, procedures, and basic human rights upon which democratic societies are build thus representing an act of oppression and injustice.

The second point that needs to be reiterated and emphasized is that the internal system of the Nazi concentration camps and the structural aspects of its prisoner society were intentionally designed by the Nazis in a way that would foster rivalry, power imbalances and conflict amid its prisoners and would make certain groups of prisoners complicit in the functioning of the whole system by thrusting them in a position where an individual's survival often depended on treating his fellow prisoners badly from a position of power.

I assume many readers here will be familiar with this chart of different categories of Concentration camp prisoners. I have gone a little bit into the problem with this chart in this answer but the problem generally is that we know little about how closely and stringently it was followed throughout the years and only have ample evidence for the widespread use of red and green triangles and the Star of David while lacking concrete evidence for the widespread use of other symbols. But the point of the chart is not so much how accurate it was and more that it represents the institutionalization of something that had been very present from the very inception of the German concentration camp: the classification of prisoners into different groups and according to this group classification different privileges and access to positions of power within the camp hierarchy.

There are numerous examples of this but one of the most illustrative is the status of Jehovah's Witnesses and other so-called "Bible-students" in Concentration Camps. I go into this more in this answer but the gist of it is that despite the relatively small size of this group of prisoners, they were after a certain point over-represented in positions such as so-called Kapos (prisoner foremen of CC work details), prisoners working in the camp administration like accountants, running the magazine or kitchen and so forth because the Nazis realized that JWs and other Bible students would not try to escape the camp.

Even if given the chance, the Bible-Student prisoners would not escape the camps. In their faith, imprisonment in the camps was basically a test from God for the coming end times and while they refused any cooperation with the war effort, they also refused to flee. Theological debates took place in the cams surrounding this in illegally printed watchtower leaflets. Anyways, once the camp administration figured this out, they used the Bible-Students as preferred prisoners for work details outside the camps. In Austria, one such detail e.g. fixed up a farm house for a high ranking Nazi where the prisoners also slept in said farm house with only one guard. They could have overwhelmed him but stayed put.

Another example of this a colleague of mine discusses in a forth-coming article on the subject of "Yugoslav" as a prisoner category is that people from Yugoslavia imprisoned in the camps had a higher chance of survival when classified as "Slovenes" or "Croats" or even "Serbs" than when classified as "Yugoslav" since "Yugoslav" still carried a stronger implication of being opposed to Nazism due to the history between Yugoslavia and Nazi Germany than the individual nationalities carried and people classified as Yugoslavs suffered stronger discrimination and higher violence from the camp administration than people classified as "Slovene" etc.

What resulting from these classifications and from the general structure the camp administration set up was a distinct hirarchy among prisoners. Historian, former political Auschwitz prisoner turned Nazi hunter Hermann Langbein compiled a book about the camp experience called Menschen in Auschwitz (Humans in Auschwitz) in which he extensively relies on prisoner testimony to re-construct a comprehensive picture of life in the camp and where he describes this hierarchy:

If one survived the initial hard times after arriving in the camp, if one had learned the rules of camp life and had made contacts with other prisoners, one became an "old prisoner". Demonstrative harshness and brutalization were often the signs of "old prisoners". This type was stronger developed in camps with a longstanding German prisoner population and thus never fully developed in Auschwitz.

An unwritten law in all camps said that new arrivals would receive harsher physical work while prisoners with low numbers would receive a good work detail. The SS didn't even have to monitor this rule too closely for it was a respected law of prisoner hierarchy. With a good work detail came a better living space and block that was not as crowded, more food, and less physical violence.

Equally as important was that a better work detail and a lower prisoner number meant better clothing. The moniker "clothes make the person" was nowhere as true as in the Concentration Camp as the episode of George Wellers illustrates:

Weller, an orderly in the Monowitz sick bay and wanted to help a friend who was still at the lowest rung of the prisoner ladder. Dressed with care he visited him in his block. He played the big cheese, smoke demonstratively despsite this being forbidden to the average prisoner, stood in the middle of block floor so that everybody had to squeeze past him mumbling apologies, in short, he did everything so that the people in the block knew how well connected his friend was.; because he knew that in their eyes the luxury he radiated as a prominent prisoner would reflect back on his friend.

(...)

All too often the old prisoners looked down on new arrivals and those who were in the worst shape in the constant struggle for survival that was the camp. Primo Levi said about this: "The social structure of the camp was build on the fact that those with privileges oppress those without them." Bruno Bettelheim wrote: "The fear of sinking into the sub-human class of prisoners was a strong motivation for waging a class war against them."