Fashion moves reliably in cycles of twenty years. It's entirely expected that 1998's visual styling looks fresh again in 2018, always filtered through an ironic-nostalgic lens of varying density.
To me, the Yellow site doesn't really look like an old website at all because HTML's creative possibilities have changed — typography, layout options, etc.
However it does look like a lot like print design from circa 1994-98. Wide font variants for all copy, strong contrast with primary colors, and so on.
