As of Jan '20, the zathura official documentation is being reworked and does not contain anything related to usage.
Looking at the man docs, it seems that zathura does not come with select / copy keybindings that mimic Vim's visual mode. The closest that comes to it is scrolling with h, k, j, l. Here's the full list of mouse and key bindings.
Feel free to open a feature request.
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