Is one allowed to play Go with a notebook?
I can only answer the Go perspective. I hear it is pretty much banned in chess.

Paper notebooks, or preferable kifus, are always allowed. If is not uncommon for tournament directors to supply players with kifus for free or cheap.

I've often seen people use electronic notebooks in Go tournaments to record their matches. However, this is usually accompanied by asking the opponent beforehand for permission (it would simply be rude not to ask). To my knowledge, this is almost always approved, in particular when the display is visible to both players during interaction. Similarly, there is not usually a problem in photographing the board. There have been no real incidents with people cheating yet.

I have never seen it in high level matches, probably simply because anyone beyond low dan level can memorize their matches to a high degree anyway. I usually record my matches after they finished, and possibly take a photo if a strange move happens which is difficult to remember.

In Go, the tournaments with large price money are exclusive to few top players, and Go bots are not able to compete with them, so there is no incentive to use them. This may slowly change in the next several years.