Time is a continuous variable, and kids know this pretty early on: they will happily tell you they are 5 and a half years old. Their second year starts when they are aged 1.0.
English in general makes this distinction: "He is fewer than 6 years old" is simply a mistake, while "less than 6 years old" is fine. (I think some dialects use "fewer" less often, in general, but would be pretty surprised if any prefer it here.)
To be pedantic, it's the first annual celebration of your birthday.
Other languages use a less confusing term for it: anniversaire (French) is just anniversary, verjaardag (Dutch) roughly means year-rollover-day, and cumpleaños (Spanish) signifies completion of another year.
I just realized something: anniversaire has the same root as the other examples I gave. It comes from the latin words annus and verso, which mean "year" and "to turn over" respectively.
late 14c., from Old English byrddæg, "anniversary or celebration of one's birth" (at first usually a king or saint); see birth (n.) + day. Meaning "day on which one is born" is from 1570s. Birthnight is attested from 1620s.
https://www.etymonline.com/word/birthday#etymonline_v_27158
In korea, they'd be age 2. There (and possibly other east asian nations?) count "age" in terms of which year you're in. 1st year, 2nd year, 3rd year, etc. The way americans do school year, except it starts at birth.
When my korean friend first told me this, I honestly thought they were messing with me. I still struggle to make sense of it. Must be a helluva birthday party on Jan 1st tho
I know there are studies on how birth date affects future success in school (though IIRC I read in Gladwell, and I'm wary of accurate his read on things are), I wonder if that effect is compounded with this system?
What we call “age” is merely the number of the most recent birth anniversary we’ve passed. That makes sense. While your take isn’t invalid, it sets up a less intuitive construct that isn’t aligned with the common meaning of the word.