돌잔치 refers to the specific celebratory practices that Koreans do for a child's first birthday, and doesn't refer to a particular first birthday in general. In English, we have the word cornerstone. It could literally mean a stone at the corner (used in architecture for the most important stone of an arch), or it could figuratively mean "foundational".
It's not from a Sinitic character at all, so not the one that means cornerstone, but from a native term for yearly anniversary. As annum is Latin for a year, even the English term could mean one (singular) year as well and hence the first in particular. Like ippai in Japanese -- literally one cupful but it generally means a whole lot more than one.