The boards are made from kaya wood. The thicker boards come from old growth
kaya, which is now protected due to overcutting in past decades. This
explains -in part- the high prices of new thick boards... the finest
examples can cost above $19K. I've seen handmade sets of shogi pieces
(satsuma boxwood, spectacular grain, raised lettering) for $3500... just
the pieces, no board. Weirdly enough, Shogi boards don't seem to have much
value on the second-hand market. I looked into those because shogi
interests me.

The ink is urushi. It's the dyed sap of poison sumac (think poison ivy/oak
on steroids), which only cures in a high humidity environment.

The burnisher is called an uzukuri, it's similar to a polissoir but comes
in 3 grades for woodworking: coarse (karukaya grass), medium (hemp fiber)
and fine (horse hair). The one in the video is probably a karukaya uzukuri.
It's often used with ibota powder (shellac wax). A looser version of the
karukaya uzukuri is a traditional brush used to scrub cooking pots. The
horse hair uzukuri is also used to prepare the Japanese copper alloys
(shibuichi and shakudo) before patination. The Roubo-style polissoir
corresponds to the roughest grade of karukaya grass.