First, the title. Cliff Hanger absolutely sounds like the kind of thing an American company would come up with to sell Lupin to the States without using the name (there's a legal history there with the estate of Maurice Leblanc, author of the Arsène Lupin stories, which is quite beyond our jurisdiction) but apparently it was not made up by Stern. Discotek producer Brady Hartel found a single frame of the Cliff Hanger logo from the Streamline dub of Lupin III's Greatest Capers, a 1995 VHS release compiling a few episodes of the second anime series. He goes on to mention that Cliff Hanger was a name attached to the original 1980 English subtitled release of The Castle of Cagliostro, a script of which was found when assembling materials for Discotek's The Castle of Cagliostro Blu-ray release. TMS apparently made this in either 1979 or 1980, predating the Stern arcade game by at least 3 years, and the logo happened to be among the materials sent to Streamline. Anime history is weird!
Next, let's move on to the English dub present here, as there's a couple of amusing things about it. The first thing to talk about is how, at this point in time, there was no English dub of Lupin III: The Castle of Cagliostro available for Stern to use, only that TMS subtitled version which of course had no English voice track. As a result, Stern mostly dubbed in their own dialogue, apparently using just anyone around the office (none of the voices are credited, sadly, because I want to know how many cigarettes the voice actor for Jeff / Jigen smoked daily to get that gravelly voice) but there are a few bits left in from the original Japanese dialogue. Specifically, Draco / Cagliostro tells his lackey to fire but says the Japanese word, "Ute", a few times you hear Cliff / Lupin call out to Clarisse, but this audio is not rerecorded- you can tell from the pronunciation of her name as 'Clarissu'- and scenes like Cliff / Lupin making his way across the rooftop and fighting Count Draco / Cagliostro on the clock gears use grunts and shouts from the original Japanese voice track. However, The Mystery of Mamo did have an English dub in 1979, but there were a few lines added to facilitate gameplay (including that deceitful "Take a right!" line) so only one line of dialogue that isn't shouting or laughing actually makes it into the game from there- Samurai / Goemon remarking, "Once again, I've cut a worthless object" which is sort-of his catchphrase in the source material (I'm pretty sure Lupin's line at the end of the scene is done by the Stern voice actor, though, but I couldn't obtain a copy of the Toho dub to check).
Nearly done now! It shouldn't surprise you to learn there's no home version of the game- even if they'd opted for, say, an MSX conversion like with Time Gal and Badlands, the fact that it's all just licensed footage would've made it a logistical nightmare and probably not worth it, especially as Stern were not in the video game business for much longer after Cliff Hanger. However, there was an attempt to bring Cliff Hanger home... Sort-of. In a podcast interview with Anime News Network in 2015 after the release of their Blu-ray edition of The Castle of Cagliostro, at 1:01:00 Discotek reps Brady Hartel and Reed Nelson mention plans to include a full playthrough of Cliff Hanger on the release by a previous world record holder on the game with a commentary track about the game (contrary to what other places on the internet say, it doesn't sound like they were planning on having the game actually playable) which would've been amazing! Unfortunately, despite their best efforts, they were unable to locate the contracts made for the game and so had to drop plans to include the playthrough.