Germany as a nation state was only founded in 1871 (after a long lead-up), sixty years after Kleist’s death, but he lived in a time that is considered formative for the idea of a German identity.

Kleist and many of his contemporaries (Schiller, Goethe, the Grimms, Eichendorff, Hölderlin, etc.) make up the foundation of the German literary canon (for better and worse) - and have been used as sources for what that German identity is supposed to be ever since. People still call Germany the “Land der Dichter und Denker” (land of poets and thinkers) which is a phrase going back to Kleist’s time.

So, he’s “very German” in the sense that the idea of “German” didn’t really exist before the 19th century and he was part of the gestation of that identity. Kleist himself was very much a nationalist in a time when much of what would become Germany was under Napoleonic rule and wrote several patriotic poems, dramas and novellas. Unfortunately, the Nazis found it very easy to misuse them for their own purposes while downplaying their liberal and republican (both not in the modern/American sense) contents.

I recommend “Michael Kohlhaas” which was also one of Kafka’s favorites.