Morphism of schemes and closed points

Question

I apologize for the vagueness of my question.
In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the "coordinate" descriptions.
For example, on page 300, the Segre embedding is described by $$\mathbb{P}_A^m\times_A\mathbb{P}_A^n\to \mathbb{P}_A^{mn+m+m}$$ sending $$([x_0:\cdots:x_m],[y_0:\cdots:y_n])\to [z_{00}:\cdots:z_{mn}]= [x_0y_0:\cdots:x_my_n]$$ I think this is because we are allowed to do so in the affine case (as the projective morphism here comes from glueing).
Usually, such coordinates are identified with its corresponding maximal ideals. However, a scheme usually has more points than those closed ones. And the set of closed points is not usually dense $(\operatorname{Spec}k[x]_{(x)}$, even though it is not locally of finite type).
Even if it is a dense subset, with Zariski topology, I am not sure if one can recover a scheme morphism by its value on a denset subset. So why is this description enough?
Any help is appreciated! Thank you very much.