By the definitions, these two can be different. E.g. $(-1)^n$ and $\{-1,1\}$, where $1$ is an accumulation point for the sequence but not for the set. Then it becomes a bit messy.
Is there any way to define so that we only have one definition contain both of the two cases? Probably, by product topology?
You could probably do it using multisets. We can define accumulation points analogously to sets: A point is an accumulation point of the multiset if, for every open interval containing the point, the sum of the multiplicities of the points in the multiset contained in the interval is infinite.