With the web, the convention right now is to treat the subdomain as a different security origin (with the exception of www). So the link should show c0ffee.surge.sh, not surge.sh.
If this is a manual setting, it probably also needs to be set for neocities.org. I noticed that wordpress.com domains were being subdomained properly.
It really shouldn't be manual, it should just always show the correct origin domain.