Why is a quantum state a vector and not a point in Hilbert space?

Answer 1

Because a state can change. That change is measured in one dimension. Vectors are one dimensional. For ex., excitability is a state that can be measured in either direction, but not sideways.

Answer 2

All Hilbert space can be represented as extensions of basic separable Hilbert space. And Hilbert space is simply Euclidean (or other) space transformed to conform to certain rules of geometry. [See source.]

One of these, in Hilbert space, we have the fundamental which connotes the dot product orthogonality relationship. To fit into Hilbert space and have that characteristic, a state must be described as a vector (x). And, as a result, the dot product of the state would be a scalar...your point.