You can think of faithfully as someone standing exactly where they said they would stand, and still being there when you come back much later. What they hold on to changes from use to use: a person, a habit, or the exact shape of an original. The picture underneath stays the same one — nothing drifted, nothing was swapped out.
From the word’s history
Wiktionary records faithfully as Middle English feithfulli, that is faithful plus -ly, built on faith. Keeping faith with something is the idea that still sits under every modern use.
Wiktionary 記錄 faithfully 來自中古英語 feithfulli，也就是 faithful 加上 -ly，字根是 faith（信）。今天不管守的是一個人、一個習慣，還是原作的樣子，底下都是同一件事：守住那份信。這是概念上的連結，說明它今天的用法為什麼能收在一起。
Wiktionary (kaikki extract, source_ref kaikki:3f6f69a190036ae4:1025563559:8807): From Middle English feithfulli, equivalent to faithful + -ly.
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