I realize the component of one vector $\vec{a}$ in the direction of the other vector $\vec{b}$ is essentially a projection of $\vec{a}$ onto $\vec{b}$ and I understand how it is calculated. What I don't understand is why exactly we call this projection a "component" in this case. If we take vector $\vec{a}\space(5, 6)$ in $2D$ space with origin in $(0, 0)$, what I am used to call the components of such vector are vectors $\vec{b}\space(5, 0)$, and $\vec{c}\space(0, 6)$. Or, in simple words, such vector has $5$ units towards $X$ direction, and $6$ untis towards $Y$ direction.
Now, looking at this picture projection of $\vec{a}$ onto $\vec{b}$ we still call the resulting projection of $\vec{a}$ onto $\vec{b}\space$ a "component". However this component does not have anything to do with components of vector $\vec{a}$ that are along $X$ or $Y$ axis. I would need some intuition on why we call the resulting projection of one vector onto another a component. It's more intuitive for me to just call it a "projection", not a "component".