You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:
If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.
The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.