Is a characteristic polynomial we consider in Linear Algebra a polynomial or a polynomial function?

Nice question! In many cases, that distinction is irrelevant, but in some cases it matters. And, when it matters, you are not right: it is a polynomial, not a polynomial function. For instance, polynomials have degrees, whereas polynomial functions don't (for instance, over $\mathbb F_2$ the polynomial function $x\mapsto x^2+x$ is the null function, but the polynomial $x^2+x$ still has degree $2$, whereas the null polynomial still has degree $0$). And the degree of the characteristic polynomial of a $n\times n$ matrix is $n$.

The characteristic polynomial of $T$ (either a matrix or a linear transformation, depending on your preference) is a polynomial, not a function. What we really care about are its coefficients. For instance, the leading coefficient is always $1$ (so that is boring) but the degree of the polynomial is the dimension of the ambient vector space. The next coefficient is (up to a sign) the trace of $T$. The free coefficient is the determinant. The other coefficients also have meaning directly expressed in $T$. All of this will be lost if you considered the polynomial merely as a function since over certain fields this process destroys the coefficients.