These things tend to get written in an imprecise way because writing things precisely is cumbersome and unnecessary once you know what's going on. But it's useful when learning to write things overly precisely so let's do that.
Most mathematical objects are actually a few pieces of information bundled together. We'll write the precise definition of the objects in question as a tuple:
Definition: A vector space over the field $F$ is a triple $(V,+,\:\cdot\:)$ where
- $V$ is a set,
- + is a function $V\times V\to V$, written $(v,w)\mapsto v+w$,
- $\:\cdot\:$ is a function $F\times V\to V$, written $(c,v)\mapsto c\cdot v$,
satisfying conditions [insert vector space axioms here].
It's important to realize that the addition and scalar multiplication functions are part of the data of a vector space $V$ itself. If you change the addition then you've changed the vector space, for instance.
Now let's define an inner product space overly precisely:
Definition: An inner product space over $F$ is a 4-tuple $(V,+,\:\cdot\:,\:\langle\:-\:,-\:\rangle\:)$ such that
- $V$ is a set,
- + is a function $V\times V\to V$, written $(v,w)\mapsto v+w$,
- $\:\cdot\:$ is a function $F\times V\to V$, written $(c,v)\mapsto c\cdot v$,
- $\langle-,-\rangle$ is a function $V\times V\to F$, written $(v,w)\mapsto\langle v,w\rangle\in F$,
satisfying conditions [insert vector space axioms and also the inner product axioms].
Notice the pieces 1-3 here are just the "data" of a vector space. So we can simplify the definition here as follows
Definition: An inner product space over $F$ is a 4-tuple $(V,+,\:\cdot\:,\:\langle\:-\:,-\:\rangle\:)$ such that
- $(V,+,\:\cdot\:)$ is a vector space,
- $\langle-,-\rangle$ is a function $V\times V\to F$ satisfying [the inner product axioms]
which I imagine is closer to the definition you've come across, but they probably just said something like "where $V$ is a vector space" instead of being precise about this being a tuple of objects.
So what you see here is that if we are careful about thinking about the functions defining a mathematical structure as pieces of data for the structure, then going from a vector space to an inner product space involves adding a piece of data, i.e. the data of the inner product function.
