Interesting question.
Let’s be very clear, HYDRA in the comics is a Nazi organization through and through. Decades after WWII, HYDRA is working to awaken Nazi robots that will conquer the globe in the name of the Fourth Reich, or to destroy it in revenge for the downfall of the Third. The Red Skull is very explicit about his ideology – he is fundamentally opposed to human equality and believes entirely in the doctrine of Aryan racial supremacy. (Just wait until I get to the comics of the 1970s where the Red Skull starts freaking out about interracial relationships) Likewise, Strucker is very explicit that he is in no ways a former Nazi and very much an active fascist.
In the MCU, the serial numbers have been filed off a bit, largely so that they don’t run into problems with ratings boards in countries that are very sensitive to depictions of Nazi symbols, but I would argue not much has changed. HYDRA believes in the total subordination of the individual to the group – hence the whole “cut off one head and two shall grow in its place.” They view democracy as inherently chaotic and prefer a political system where order is imposed from above through force – hence, Alexander Pierce’s entire worldview. And they generally are governed by a single charismatic individual.