The gameplay is intrinsically tied up in the storyline.
In Duke Nukem 3D, the environments, NPC characters and overall production design are relatively 'realistic', which is contrasted with Duke's superhuman run-and-gun ability and total lack of psychological complexity to emphasize that, yes, this guy is a cartoon like Bugs Bunny - but also a transgressive and rebellious figure because of that cartoonishness. The gameplay in DN3D is a huge component of Duke's characterization.
The strippers in DN3D are just women who strip. They didn't shriek innuendos at you at all. They didn't declare themselves stupid in scripted sequences. They did their jobs in exchange for money. Again, the interactive world was 'realistic' (for a 1996 game) and Duke was the cartoon avenging force who flew down to Earth to right the wrongs. Kinda.
(People have noted that in Duke II, Duke is a full-fledged celebrity. But that's part of the 80s action parody - Duke is simultaneously an everyman and a celebrity, a blue-collar Joe and a relentless killing machine. But in the grand scheme of things, he's far more populist hero than elitist billionaire. Think of the Ghostbusters - beloved by the public despite being impoverished, uncouth and totally reckless. Duke is a similar underdog. The aliens in Duke II interrupt his TV appearance and steal his moment in the spotlight, because Duke just can't catch a break!)
BeanTaco hit it right on the head: Duke in DNF is clearly an entirely different character, to the point of being essentially the opposite character. He's an impostor.
With the 2-weapon thing and the slower movement, Fake-Duke is 'realistic'* while the world is cartoonishly exaggerated. Women are literally subhuman cock-hungry idiots and not just perceived that way by a flawed protagonist. The president is a liberal pantywaist caricature who (allegedly) gets his comeuppance when Fake-Duke "proves" that the world does need another George W. Bush. Y'know, someone who's motivated by sheer literal prejudice - we don't need no evidence of WMDs. Those alien Iraqis are obviously going to rape our (white) women because, naturally, that's what they do.
It bears repeating that the "normal" people in these games represent you, the Duke's Bro. Those moronic groupies sucking Fake-Duke's cock and getting raped are you.
Fig. 1: Your Face Here.
These NPCs are so ridiculously idiotic and annoying that Duke actually appears rational, levelheaded and intelligent by comparison. The first hour or so of the game is spent with Fake-Duke just walking around gawking at the absurdity around him, because he's normal now. DNF is almost evangelical about this. If you're not like Duke, the game says, you're a fucking moron and a bitch.
Even DNF's cover shows Duke, from the perspective of a kneeling person, holding his gun like a cock. The game's box is telling you to suck Duke's dick. I guess it's truth-in-advertising.
Fig. 2: John Romero's about to make you his bitch.
Of course, that's not the real Duke. Duke fought the pigs, not his fellow man. Plus, the point of DN3D and earlier games, to an extent, is that no-one can be Duke. He's an amoral force of nature. He's a superhuman wish-fulfillment character. He's a cartoon.
*The DNF health system, bizarrely, goes in the opposite direction, replacing reasonably plausible sci-fi health pickups with the purely fantastic 'ego' system. So while Duke3D is a mortal (every)man who transcends his situation through impossible skills and bravado, Fake Duke is immortal despite lacking much remarkable talent or courage.
