The black album is probably the album that got the most people into guitar until maybe Taylor Swift, and was big enough to get spoofed by Weird Al twice (one full cover, and once in one if his medlies). If that doesn't make them relevant in the 90s, I don't know what does.
If you mean specifically to the metal community, then I would say the black album still holds its own, but that was the very beginning of the 90s...and they've tapered off since then, with a blip for Death Magnetic, I think.
I think it might be easy to look at the major studio albums from Metallica throughout the 90's and 00's and note the decline in quality and/or popularity. And maybe that's what happened with the comment from @DoctorStoner that made me WTF out loud. But, for those of us who were alive and aware of music trends in 1991, the black album is unquestionably influential. 33 million copies sold. None of the other bands mentioned in @DoctorStoner 's post had an album that popular in the 1990's, and, in fact, the only two albums by those bands that top the black album are Led Zeppelin [IV] and Back in Black. I'm pretty sure you could add up all of Black Sabbath's album sales over their entire career and not top the black album's numbers.
But after the black album, Load performed okay and Reload performed okay, and then it was downhill sharply after that. But Metallica didn't really release that much new material in the 1990's. Three regular albums, one cover album, one symphonic album. But one of the best selling cover albums and one of the best selling symphonic albums to date... But the black album put Metallica on the mainstream music map. Justice was considered a smash hit for a metal album, and it sold 11 M copies, >8 M of which sold after the release of the black album. So...without the black album, Metallica would have been a popular metal band, but, the important point here is that metal would have been a lot less popular of a genre. Certainly the big boom in interest in Megadeth and Anthrax can be at least partially, if not mostly, attributed to the massive success of the black album.
Growing up in the midwest, and not being particularly aware of metal music in 1988, I never really knew much about Metallica until Justice. That album was big, but just think about releasing an album that was big, and then having the next one be more than ten times bigger.
And the parodies didn't stop with Weird Al Yankovic. Tons of sketch comedy shows spoofed Metallica in the early 90's. For instance:
But, anecdotally, tons and tons of my friends discovered metal music directly because of the black album.
And when Load came out, it was possibly the most anticipated album ever. I remember when the video for "Until It Sleeps" debuted on MTv and pretty much everyone thought it was just weird and hoped that the rest of the album was better. I mean, it sort of was, but it also sort of wasn't. It was disappointing enough that it seemed like only the most optimistic Metallica fans were excited for Reload. I think that maybe if they had just called that album Shit Sandwich, it would have generated more buzz. But, despite that, "Fuel" and "2x4" and "Hero of the Day" and a few other songs still made enough people happy that they kept doing tons of shows and making tons of money.