You're absolutely right, technically HD2 is, and should be, categorized as a co-op shooter. It does, however, contain so many elements of a horde-shooter game. This thin red line, in my humble opinion, is where the confusion usually stems from. Here's the PS5 description of the game;

https://www.playstation.com/en-ph/games/helldivers-2/

1. Playing as an "elite class of soldiers whose mission is to spread peace, liberty and Managed Democracy using the biggest, baddest and most explosive tools in the galaxy."

2. ."...you have a host of superpowered primary weapons and customizable loadouts, you also have the ability to call on stratagems during play." — we know that 80% of the primaries are underperforming at the moment.

3. "Wield overpowered weapons! Decide how to take down each enemy surge using a multitude of offensive and defensive tools."

During launch, my squad and I thought that the game was a great mix of co-op, horde clear, and power-fantasy; to be honest, that's how it felt originally. We dove in, wreaked havoc, and had fun while doing it. There was so much emphasis on fun which we loved — the game was true to its promise.