As you may have noticed, there are a lot of movies with Nazis, but not a lot of movies about a Nazi. There's no Otto the Lonely Nazi or Hartmut and Wilhelm Go to Camp.
Casablanca is clearly not about Major Heinrich Strasser, but he's far more fleshed out than some other Nazis in films, certainly among films made in that time period. He isn't just a two-dimensional stereotype, barking out orders to his underlings in a harsh German accent and then walking down the street, shooting nuns and kittens as he goes.
He's nasty, make no mistake about it, but there's a human element to him. He's usually minimally civil to the people around him. When he's speaking casually to Renault (whom he mostly trusts) and Rick (well, whom he trusts more than Laszlo, anyway), he almost seems like your run-of-the-mill jerkface, rather than an insane killing machine. For other movie Nazis (we're looking at you, SS-Standartenführer Hans Landa) a nightclub full of patrons singing "La Marseillaise" in 1941 at the top of their lungs might justify machine-gunning the whole lot of them without so much as a second thought. Strasser just steams and orders Captain Renault to close down the club.
(Note: he is still very much a Nazi who murders people in cold blood but it's interesting to see that one of the more nuanced portrayals of a Nazi officer came from a movie shot and released during WWII.)
Of course, he gets his in the end. He goes out with a bang…if not with an Oscar-worthy death scene.