Why do we use the word "scalar" and not "number" in Linear Algebra?

So first of all, "integer" would not be adequate; vector spaces have fields of scalars and the integers are not a field. "Number" would be adequate in the common cases (where the field is $\mathbb{R}$ or $\mathbb{C}$ or some other subfield of $\mathbb{C}$), but even in those cases, "scalar" is better for the following reason. We can identify $c$ in the base field with the function $*_c : V \to V,*_c(v)=cv$. Especially when the field is $\mathbb{R}$, you can see that geometrically, this function acts on the space by "scaling" a vector (stretching or contracting it and possibly reflecting it). Thus the role of the scalars is to scale the vectors, and the word "scalar" hints us toward this way of thinking about it.

Not all fields are fields of numbers. For instance, it makes sense to talk about vector spaces over the field of rational functions $\mathbb R(X)$ but the scalars in this case are definitely not numbers.

Scalar gives you a sense of what the "number" does. A scalar scales a vector, stretching or contracting each of its coordinates by the same amount.

While yes it is a number in common parlance (as long as you are working over a field of numbers, which you probably are), in the context of linear algebra, numbers really just serve this purpose (unless you are in one dimension, in which case vectors $\textit{are}$ numbers and there is some ambiguity).