> And are there specific vector spaces where one ANN is preferred? Or will there be some universal approach that just works for everything?

This is a great question for research right now. Basically euclidean (and inner product) search is quite well understood, but for distance measures such as edit distance we still have no idea what the best approach is.

The most general result is https://ieeexplore.ieee.org/abstract/document/8555102 , which works for all normed spaces.

However it is possible that people will try to just map everything to inner products using neural networks.