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.
TreasuryDirect also feels classic. If you're not familiar, it's a US government website to buy and sell certain types of treasury bonds. Some great features:
an image so you know you didn't typo your username (haven't seen that in well over a decade)
clicking a link is a new page, and clicking back breaks stuff and makes you login again
until recently, you couldn't paste in the password field
It does do some modern-ish things with page layout, but not that modern, like maybe early 2000s modern. But it's perennially stuck about 20 years in the past.