Use of word "axiom" in definition of vector spaces
Axioms in modern mathematics means the same as a set of properties, or conditions. Different things may or may not satisfy them. When you define a vector space you are basically saying that a vector space is anything which satisfies those axioms. The axioms are not intended to have any deeper meaning than that. They simply single out some things among all by listing properties. It's simply a definition. In the olden days, some philosophers and teachers insisted on attaching some vague unverifiable truth to the notion of axiom. That is long gone (or should be).
My understanding of axioms is that they are base assumptions which are taken to be true.
In some sense, that is true here. They are the base assumptions you are allowed to make when someone gives you a tuple $(V,+,0,-,\cdot)$ and tells you "this is a vector space". If you know some first order logic: you could make the language of real vector spaces, which has function symbols $+,-,0$ and for each real number $r$ a function symbol $r\cdot$. Then the above axioms are all formulatable in the this language. (Edit: and you need to include the theory of the real numbers as well, which is left implicit here.) The models of this set of axioms are vector spaces; and to prove that something is a vector space, you prove that it satisfies those axioms.
Yet from this definition, it's necessary to show that the axioms are "satisfied" for a specific set in order to conclude that the set is a vector space. Is that somehow different than "proving" the axioms are true for the given set?
Not really, except for the very subtle difference that something can be true without being provable. If someone asks you "is this tuple a vector space", your only recourse is showing that all the axioms hold, or that one of them does not.
The axioms are defined to be true for vector spaces.
When you are trying to show that something is a vector space, you are in fact verifying (not "proving") that the axioms hold.
It is a bit of wordplay here.