Is there a way to overlay two similar UV islands with identical vertex count? Sometimes UV unwrap has several nearly identica islands. However, more often than not they differ a little. If I want to keep identical texture for these -- is there a good way to overlap them perfectly? I know I can do it vertex by vertex, but I hope there is a better way.

I know I can do it for the whole model if it is symmetrical with Symmetrize option from mesh menu. I am talking about selected islands, not all of them.
Automatically unwrapping  identical geometry pieces one by one should produce identical UV layouts.
I.e. if you have a row of identical parts of geometry, select one, do unwrap, select another, etc. Selecting all at once may produce alike, but different islands.

If geometry is not identical, I'm afraid you're stuck with manually scaling down to zero required vertexes.