Yes, the claims are pretty much in the same spirit. Although the first (Weierstrass's) theorem [1] was stated for real-valued functions in a 1-D closed interval [a, b], Stone-Weirstrass is a generalisation of the above theorem [2] that's applicable in more general scenarios. Here is the formal statement:

- [1] http://mathworld.wolfram.com/WeierstrassApproximationTheorem...

- [2] http://mathworld.wolfram.com/Stone-WeierstrassTheorem.html

Neural Networks use a different "basis" (sigmoid, ReLU, etc.), but the underlying idea shares the same spirit.