Where does the base come from in a subsequence count?
You can think of a subsequence $Y$ of $X$ as a characteristic vector $\chi_Y$  that has $m$ elements, one per element of $X$. The $i$-th element of $\chi_Y$ is $1$ if the $i$-th element of $X$ is also in $Y$ and $0$ otherwise.

It is easy to see that this is a bijection: every subsequence $Y$ has a distinct characteristic vector $\chi_Y$, and every characteristic vector $\chi_Y$ is associated to a distinct sequence $Y$.

This means that the number of subsequences $Y$ is equal to the number of vectors $\chi_Y$. Since each vector $\chi_Y$ has $m$ elements, each of which can have one of $2$ possible values, this number is $2^m$.

A concrete example of a sequence $X$, a subsequence $Y$, and its characteristic vector $\chi_Y$ is the following: