I mean, judging from what I have read on this subject, the economy they operated begins with moderate socialist interference and ends up full scale authoritarian socialism by the fall of the regime. The state was the economy and operated all the major industries by fixing market conditions or outright seizing businesses. There was no corporatist/statist economy like all of the other fascist regimes, and the economic structure had more in common with Soviet Russia than it did with fascist Italy.
Furthermore, the party itself claimed to be the National Socialist German Workers party (sometimes called just "the workers" party internally, or "the Reich"). Which is essentially the group, themselves, calling out their ideals.
This is also not to mention the fact that Stalin and Hitler were very staunch allies, sharing political views, transferring war supplies, and trading information right up until operation Barbarossa when Hitler attacked Russia. Even then, Stalin could not believe he was being betrayed due to the nature of the closeness of their relationship prior to those events.
Meanwhile, Mussolini was an association of convenience, and Italy needed Germany more than Germany needed Italy.
However, many sources claim the regime was fascist, in spite of there being numerous ideals, structures, policies, and laws that point to this categorization as not being the case.
Can anyone give me a legitimate answer as to why?
While you’re waiting for an answer to your question, you might be interested in the section of our FAQ on how socialist was National Socialism, with prior answers by u/kieslowskifan and u/g0dwinslawyer.
Authoritarianism has very little to do with delineation of right-left politics. As you have pointed out, both Nazism and Soviet Communism were authoritarian but on opposite ends of the spectrum. There were even right wing anarchists like the NRF, and of course your standard gamut of anarcho-Communists. Thus established, authoritarianism, structure, warmongering, and state-run economies have little to do with the left-right spectrum.
A better gauge is the purpose of the state. Nazi nationalism as we are all well-schooled on, focussed on the SUPERIORITY of the Aryan race and the INFERIORITY of Slavs/Jews. In this way, Nazis like other Fascists sought the expansion and welfare of their own particular demographic at the expense of others. The latter part is crucial, fascism and the right-wing trumpets the primacy of some form of inherited purity which creates superior unity/strength.
Soviet Communism was very much the reverse, it focussed on the EQUALITY of all mankind. Whether this translated into reality on an individual level points to the success/failure of such an objective, but generally Soviet totalitarianism was predicated on squashing forms of inequality, for better or for worse. Communism and the far-left hence trumpets the primacy of some form of learnt moral ideology which creates unity/strength. Crucial to this is the idea of reform and rehabilitation, which are absent in the far-right.
These differences often translated to similar policies but more often than not, led to different outcomes. For instance, both considered the eradication of capitalist structures within their target demographic to be imperative. The difference can be well-observed in the follow-on policies. Capitalist Jews were present in both early Nazi Germany and USSR, and both had their assets confiscated. The Nazi approach as we all know, involved the extermination of impure non-Aryans. The Communist approach was predicated on re-education and reassimilating capitalist Jews into the Soviet society. When the last emperor of China was captured by the Communists, considerable effort and expense was made to portray his re-education and subsequent re-entry into society as a reformed worker.
You are correct in identifying early political links between Nazism and Communism. It should be noted that until the late 1930s, Nazi Germany was not any more ideologically hostile towards Communism than the rest of Europe. After all, the Soviets had just fended off an international coalition consisting of almost every country in Europe and even Japan. Moreover, the Germans did not have a colonial empire, while the British/French literally kept 65% of the world population as an underclass in their societies.