In Euclidean Space and Cartesian coordinates system, we know the vector projection of Vector u onto a vector v is simply
$$ \vec{P} = \frac {(\vec{u} . \vec{v})\ \vec{v}}{\vec{v} . \vec{v}} $$
What would be the vector projection on polar or spherical coordinates or alternate coordinate systems?
Suppose we have vectors defined as
$$ \vec{u} = u^α e_α $$ $$ \vec{v} = v^β e_β $$
Does a projection vector of u on v becomes
$$ \vec{P} = \frac{(u^α v^β e_{αβ}) \vec{v}}{v^α v^β e_{αβ}} $$
Where $ e_α $ represent the basis vector and $ e_{αβ} $ the metric tensor.
That looks correct. The dot product formula for projection doesn't depend on the coordinate system, though you need to know how the dot product actually works, which can be determined by its action on a set of basis vectors.
$$\vec u\cdot\vec v = \Big(\sum_\alpha u^\alpha\vec e_\alpha\Big)\cdot\Big(\sum_\beta v^\beta\vec e_\beta\Big)$$
$$= \sum_{\alpha,\beta} u^\alpha v^\beta(\vec e_\alpha\cdot\vec e_\beta)$$
$$= \sum_{\alpha,\beta} u^\alpha v^\beta(e_{\alpha\beta})$$
But the vectors $\vec u$ and $\vec v$ need to be in the same tangent space (based at the same point) in general. And the vectors are not points like $(r,\theta)$ in polar coordinates.