I asked on Facebook as well and it really seems to be at the discretion of the staff member.
One person mentioned:
We have the Victorian companion card and we were going to try to use it for a taxi discount in Tokyo, but the driver just gave it to us without documentation
So, you can bring the card and try to use it, and should still get the discount. But be aware there may be occasions where staff are only trained to accept the Japanese ID and may be hesitant to accept a foreign ID they may not understand. In that case, you may want to ask for their manager. If anything, it could be an opportunity to train the staff on different IDs!