If this particular story isn’t directly adapted from some story in the original LUPIN III comics, then it’s been due for a very, very long time.
Seeing as how Arsene Lupin - - the original character whom this whole dang franchise is based off of - - was something of a French superhero, it’d make a whole lot of sense for his heir to eventually run into a character riffing off the Phantom of the Opera (a contemporaneous character who was also something of a French superhero.) Hell, there was even a portion of the LEAGUE OF EXTRAORDINARY GENTLEMAN comics where the League ran into their French counterparts, les Hommes Mystérieux. Their ranks included the likes of Fantomas, the Nyctalope and, of course, Arsene Lupin, and they ran afoul the Phantom at times - - so there's proof of how much sense the idea makes!
Maybe there’s no additional relevancy to this show, but I’d gladly use this opportunity to point you lunatics in the direction of such older adventure stories that have influenced the ones you’re more familiar with. Putting the stuff you love in its greater historical context honestly makes you appreciate it more - - I swear!
Getting on the actual plot, though, I’ll tip my top hat to this show for being rather deft at the sort of misdirection that these master thieves have at their disposal. I’ll fess up to it - - this is the third time I’ve felt pretty stupid upon realizing who Fujiko was in disguise, knowing that I should’ve spotted her cover story right away.
What?! She was the understudy all along!? She’s playing me like a fiddle with the same ease that she’s playing all the guys in Lupin III’s gang?