In ACP I can create only categories for Gallery but I cannot find any section where I could see album list and manage it.

I need something like for blogs, where I can see list of blogs, but Gallery does not have any.

You can manage the albums in the frontend.

I need mass move albums. There is no way how can do it.

If you need to mass move albums from one category to another, you can click the dropdown arrow next to a category and choose to "Move/Delete Content".