You skipped the:
Which means the only way to call them is either with: nicknames, aliases/titles, add prefix/affix to it, etc. -- Kinda like how the adventurers refer to him as [Herb].
But it's still kinda iffy, since they introduce themselves with their given name (heroine did at the start); plus
1. Nicknames tend to be what close relations call you.
2. Aliases / titles would need fame, tho guess you can come up with your own.
3. Prefix/affix is more cultural -- aka mr/ms/mrs, sir, Jap's -san, -chan, -sama, etc.
Which, with only having one name, is kinda redundant(?).
Anyways, lol the clothes, most of them look modern, so no clue how they'd make it by hand (mom's supposedly a weaver).
You missed an "only" there. The quote is:
"C-calling you by name only - um... is only for very close relationships..."
She's saying that calling him without an honorific (san, sama, kun, etc.) is only for close relationships. In other words, it's not any different than Japan is today. So yeah, it's a bit of a mashup between the European style "no family name for regular people" naming scheme and Japan's "use an honorific unless it's someone you're very close to."
Medieval Japan's commoners also didn't have family names, but their situation was a little more complicated and I'm not qualified to comment on it.