In my Ukiyo-e books, for example, I see very few
prints with a border.

My question is how is this done? Are the color blocks and paper cut so
precisely that no pigment leaks up onto the baren in printing,
or is the paper printed with margins and then trimmed after, or what?

In the old days, they had no interest in having margins on the finished
item, so just cut everything as close as they wanted. There is no
'bleed' printing possible in this technique, so if fine (or no) margins
are what you need, trimming after printing is the way to go ...