Curve25519 key exchange in detail
TL;DR: The public key is not a point, it is the $x$-coordinate of the point.

The base point of the curve has been chosen to be the point $G = (9,y_0)$ with $y_0&gt;0$, and the curve Curve25519 is used in its Montgomery form given by the equation
$$
y^2 = x^3 + 486662x^2 + x.
$$
The main operation on elliptic curves is the scalar multiplication, and in this case, it is computed with the Montgomery ladder algorithm with formulas that use only the $x$-coordinate of the input point, and output the $x$-coordinate.

For one valid $x$-coordinate of a point $P$, there are two possible points, $P$ or $-P$, so if Alice private key is $a$, then her public key is $x([a]G)$ (the $x$-coordinate of the point $[a]G = G + G + \cdots + G$, the output of $\operatorname{Curve25519}(a,9)$). Since the $y$-coordinate is not used, then it corresponds either to the point $[a]G$ or $-[a]G$, then Bob can compute easily $x([ba]G)$, which corresponds to either $[ab]G$ or $-[ab]G$.

Basically, the $y$-coordinate is not useful to get a shared secret, and not useful for the computation since the formulas that do not use it on Montgomery curves are pretty efficient.