half: one of the two parts something splits into, or the point where it splits
This word comes from the idea of cutting something in two. From that one picture it does two closely related jobs, and Hebrew uses the same word for both because they are the same thing seen from two angles: the part you get when you divide, and the place where the division falls.
Most often it names a half portion. That can be half of an amount (half the blood, half a cubit), half of an object (half the altar, the child cut into two halves), or half of a group of people (the half-tribe of Manasseh, half of the workers). In every case it marks a quantity that is one of two equal or roughly equal parts.
The same word also names the middle, the halfway point of a stretch of time or space. The middle of the night is its half, the midpoint between dusk and dawn; the midst of a person's days is the halfway mark of a normal lifespan; halfway up the altar or in the middle of the sky is the point that divides the whole in two. English needs the word middle or midst here, but Hebrew is still saying the half of it.
These are not two unrelated words. The portion and the dividing point are the front and back of a single idea, which is why one Hebrew word covers ground that English splits between half and middle.