2. D.

Since $S_{2 n}=S_{n}+q^{n} S_{n}$, then $\frac{S_{n}}{S_{2 n}}=\frac{1}{1+q^{n}}$. When $|q|<1$, $\lim _{n \rightarrow+\infty} \frac{S_{n}}{S_{2 n}}=0$. Therefore, $M=\left\{0, \frac{1}{2}, 1\right\}$, so the number of subsets of $M$ is $2^{3}=8$ (subsets).
2. $S_{n}$ represents the sum of the first $n$ terms of a geometric sequence with a common ratio $q \neq-1(q \in \mathbf{R})$, and let $M=$ $\left\{x \left\lvert\, x=\lim _{n \rightarrow \infty} \frac{S_{n}}{S_{2 n}}\right.\right\}$, the number of subsets of the set $M$ is ( ).
A. 2
B. 4
C. 7
D. 8