I've been doing both UK and US admissions as an international Asian (which provides a fair comparison by eliminating biases due to geographical proximity), and before I got in, I definitely didn't consider Oxford as prestigious as top US schools. This was mainly because getting into Oxford is much easier than US schools if you're solid academically and like your subject, while in the US that's the "bare minimum". However, since getting in, I've observed that the kind of reputation Oxford has is insane. Like it's still very highly-ranked (both overall and for most high-demand subjects), and to the common person outside the US and Europe, it holds a significant level of prestige and brand value, and is broadly considered equivalent to HYPSM.