It was a passenger Zeppelin, operated by the German Zeppelin Airline Company on the transatlantic line. They just used the Lakehurst Naval Base as the landing point because it had the facilities for all the US Navy's Zeppelin's.
Its no different than Air China the national carrier of the PRC having flights to the US today.
As u/DBHT14 said, the Hidenburg was a civilian aircraft operating at at time when the two countries were at peace. The advent of transatlantic passenger-carrying airships was a technological marvel of the time and newspapers in the US often reported on the departure and arrival of these airships and their passengers.
This does not mean that the arrival of the Hindenburg or her sisters was without any political overtones. In July 1936 two members of the British Parliament charged that the Hindenburg flew over sensitive military installations in the UK and demanded that the airship fly a set pattern. City officials in Leeds also protested the Hindenburg's flightpath over their city as they felt she was photographing the city's munitions factories. A number of commentators in the US also recognized that the tour of a swastika-bedecked airship was not just a matter of travel, but a deliberate propaganda tool of the Third Reich. The group the Jewish War Veterans of the United States vigorously protested against any German-American cooperation in civil aviation. The fear of this Jewish-American pressure group was that German airship flights were a step in the development in a formal treaty system for the development of civil aviation (at this time in civil aviation, the boundary between state and private industry was very fuzzy and indistinct). Michael Weinstein, the head of the Jewish War Veterans claimed Germany's intentions for civil aviation were quite clear:
[Germany has] one underlying and motivating scheme in promoting flights on the Zeppelin Hindenburg and other airliners to this country- the propagandizing of the Nazi principles throughout the world.
Press reports in the US about the Hindenburg noted that there were usually fears of anti-Nazi demonstrations when the Zeppelins landed. Occasionally, these fears came to fruition. A red airplane with the banner "Fight Fascism- Help Free Thalmann" (a reference to Ernst Thälmann, a KPD leader jailed by the National Socialists in 1933) overflew the celebrations marking Hindenburg's 11 May 1936 arrival.
Yet the fascination with technical modernity often overshadowed the reservations about the political implications of the Zeppelins' arrival. Thousands of American visitors flocked to see the airships when they were in port and the American press detailed their construction and floated a vision of a network of fully functional transatlantic airship lines by 1940. There were limits to the appeal of technological modernity because of the Third Reich's instrumentalization of the glamor of flight. Several US commentators reported negatively on reports of the dismissal of the Zeppelin company manager Hugo Eckener on account of his decided lack of enthusiasm for the new regime. One US reporter wagged it was because the Propaganda Ministry feared a "bigger gasbag" in the form of Eckener's Zeppelins. The NYT's 10 May 1936 report on Hindenburg's arrival was a triumph not of the Hitler, but the Zeppelin's engineers:
[flying above the world] one forgets the social and political tension of Europe, the dread of war, the uncertainty of the future in this auspicious triumph of the engineers. Men still dream of enlarging their technical powers, of conquering space. All may not be morally well with the world, but it is not a world utterly lost when a Hindenburg can be built and navigated with such dramatic success.
As the NYT somewhat purple prose indicates, many people in the 1930s invested a great deal of hope and pride in the peaceful use of aviation in the name of human progress. The Third Reich's patent exploitation of aviation's symbolism was an unpleasant counter to this hope for many Americans.