Your question is a nice example for how mathematics of today works and how certain notions emerged. There is no fancy definition of a vector in mathematics, the content of the meaning what a vector has to be was shifted to other objects: Mathematicians have abstracted certain properties of "objects" that appear in geometry or physics. This fits better the axiomatic requirements of today's mathematics.

Near the end of the 19th century, a "vector" was an ordered pair (A,B) of points in an affine space. This was also called a "fixed vector", where one could imagine (A,B) to be an arrow beginning at point A and ending with its tip at point B. In today's differential geometry one finds some relicts of this situation, when a "vector" is usually given together with its base point to which it is attached.

In mechanics, there appeared so called "line-bond vectors", vectors that were considered equivalent if they differed only by a translation along the line through A and B (if A is unequal to B). "Free vectors" were vectors considered to be equivalent if they differed only by a translation in the affine space. Free vectors can represent translations. Translations can be composed and inverted - they form a group. Translations can be scaled by multiplying with a number.

From these properties emerged what is called a "vector space".
Due to the axiomatic requirements of mathematics, one puts the cart before the horse:

First, one defines - abstractly - a "vector space" (over a field (K,+,0,$\cdot$,1) ) to be a group (V,+,0) on which K acts "compatibly" via a homomorphism (of rings with unit) from the field K to the group endomorphisms of V:
(K,+,0,$\cdot$,1) $\to$ (Hom$_{Grp}$(V,V),+,0,$\circ$,id$_V$).

Afterwards, one defines a "vector" to be an element of a vector space.

So the abstracted properties appear in the definition of a vector space, not in the definition of a vector.

The original geometric content of a vector appears only later as a very special case, when a real vector space acts on a (real) affine space as its space of translations, and when these translations are depicted eg. by arrows.

(N.B.: Historically spoken, a vector was not even a pair of points, but could have different meanings, e.g. as a pair of parallel planes in 3-dimensional affine space. It also took some time till the notion of a vector space emerged and till various fields K of scalars or even skew fields were admitted. Generalizing the notion of a K-vector field from a field K to a commutative ring R with unit gives today's notion of an R-module.)