It's not obscure, but, for me, Wikipedia is the ultimate example of the old internet that still persists today.
Free to use, no account required, ad free, non-corporate, multilingual, heavily biased toward text, simple and utilitarian design. Hyperlinks concatenate relevant pieces of information, which serve as the means to navigate the site. The code is very simple (seriously, view the page source of a wikipiedia article). It's based on the human desire to learn and share knowledge with others, and has remained resilient to corruption by commercial interests that pervert that desire for monetary gain. It's a beautiful thing.
Not a website, but since you mention BBSes...one thing that would look pretty familiar to a 1990s Internet user would be most of the text-based MUDs, the ancestor of MMORPGs, that are around.
The MUD Connector is still around, and still has a list of active MUDs.
While I suspect that most dedicated MUDders use dedicated clients, the base protocol is still normally telnet, and you can use a plain old telnet client to play...a protocol that predates Internet Protocol itself.
They are trying to be 90s, but I love it. I thought they had a site counter at some point, but maybe I am misremembering and it was just the guest book.
Extremely useful website for collectors of dead media formats (LD, D-VHS, HD-DVD, CED, VHD, etc.) Still has an old style interface with priority given to function and utility over styling. Also has a storefront where you can buy and sell discs.
My healthcare services websites. Their website and mobile app require separate logins. The website logs in then redirects to a completely different website.
They have a tax-free “store” that feels like a completely different website.
Everything is laid out using what seems like the idea of middle management and not modern design philosophy.
4-ch.net (not to be confused with 4chan) is a 90s BBS that is still online and occasionally active. It's neat to see posts from the 90s still on the front page.
Sites that have old forums. There aren't many anymore, but ones I've seen that have been very helpful of late include car sites, a timeshare forum, and the Fantasy Grounds forum (my virtual tabletop of choice).
I'm sure there are others out there, but it's definitely more rare than it used to be. Is Something Awful still around?