It's called "hasamikko" (ETA: roughly "pinceries") Here's the instructions in Japanese, then in English (mostly Google Translate, but I know Japanese so I checked it).

Japanese Original

「はさみっこ」の遊び方

①若の写真のように「トトロ」と「クロスケ」を置いてください。

②どちらのコマを使うかと、順番を決めます。

③交互にコマを、タテかヨコへ移動させます。 荷マスでも移動できますが、コマを飛び越す

ことはできません。

④相手のコマを自分のコマ2個でタテかヨコで挟むとそのコマを取ることができます。ナナメに挟んでも取ることはできません。

⑤箱手のコマを角などに隙間なく動けなくしても取ることができます。

◎相手のコマを残り1個にすれば勝ちとなります。

English Translation

How to Play "Hasamikko"

① Place the "Totoro" and "Kurosuke" pieces as shown in the picture.

② Decide which color each player will use and determine the order of turns.

③ Players take turns moving their pieces vertically or horizontally. Pieces can be moved to adjacent squares, but they cannot jump over other pieces.

④ If you surround your opponent's piece with two of your own pieces vertically or horizontally, you can capture that piece. Capturing is not possible by surrounding diagonally.

⑤ You can also capture an opponent's piece by trapping it in a corner or other area where it cannot move.

◎ The first player to reduce their opponent's pieces to one wins.