Even though, algebraically, it is obvious that projecting the points of an arc vertically to the x-axis to find its length doesn't work, which postulate states that you cannot do that?
Here's an example:
You have a circle with a radius of length $r$. The length of the arc which is a fourth of the circumference of the circle has a length of $0.5\pi r$.
However, what if, instead of calculating the length like this, we project the points vertically to the x-axis, and get the length of the arc like that? Then, the length of the arc would be the radius, $r$.
Which postulate (or something else) states that you cannot do this in Euclidean geometry? Is it just for the sake of consistency with the rest of the system or something else?
It follows from the Pythagorean theorem.
Given a line segment not parallel to either axis, we can form a right triangle having the line segment as hypotenuse and whose two legs are parallel to the two axes. If $a$ is the length of the projection onto the x-axis, and $b$ is the length of the projection onto the y-axis, then the length $c$ of the segment satisfies $$c^2 = a^2 + b^2$$ If $a$ and $b$ are both non-zero, then $c > a$ and $c > b$. So projecting line segments onto an axis does not give the correct length.
For more general curves, we define their arclength by refining piecewise-linear approximations. That is, we choose a finite number of points $\{p_n\}_{n=0}^N$ in order along the curve (i.e. $p_n$ is between $p_{n-1}$ and $p_{n+1}$ for each $n$ where all three points are defined). Then we consider the sum $$\sum_{n=1}^N d(p_n, p_{n-1})$$ If sums are bounded above, that is, there is some $M$ such that for all such collections $\{p_n\}$ the sum is $\le M$, then we define the length of the curve to be the least such $M$. Otherwise we say the curve has infinite length (yes, such curves exist).
Now. if the line segments making up the piecewise approximations fail to give the right length when projected onto an axis, obviously, their sum will not be correct either. And neither will the least upper bound of the projected sums match the least upper bound of the actual sums.