Is it bad to call series a generalization of sum?

Concept $X$ generalizes concept $Y$ if every instance of $Y$ is a special case of $X$, at least with regard to some relevant property of the concepts involved.

The infinite series $\sum_{n=1}^{\infty}s_{n}$ is defined to be the limit of the partial sums of a sequence $(s_{n})$.

Given a "finite summation", it can be thought of as an infinite series where, for some $N$, we have $s_{n}=0$ whenever $n>N$, in the sense that the infinite series will converge to the value of the finite summation (this is the "relevant property" in this case).

Therefore the concept of an infinite series can be regarded as a valid generalization of the usual concept of summation, so the answer to your original question is no.

General topological spaces do not have a large number of the nice properties that metric spaces have. E.g. without the Hausdorff property a sequence in a topological space can have more than one limit (a very undesirable property). I don't think, however, that anyone would disagree with the statement that topological spaces generalise metric spaces.