Was bakelite used for computer data storage like it was used for grammophone records? My brother says he remembers this but I cannot find it on Google.
I don't think so, and I don't recall reading or hearing about any such data storage technology, although mechanical data storage (in its broadest sense) might have been done using Bakelite. For example, cam-actuated "programmers" that control electrical equipment may may have used Bakelite cams, and this could be considered to be a form of data storage. But for storing arbitrary data, Bakelite does not seem to be a suitable techology. One of the earliest forms of mechanical data storage was paper tape, which derived from telegraph (Telex) technology.
It is true that early gramophone records were made from Bakelite, but that was replaced by shellac, and then vinyl.
And people did use vinyl grammophone records to store computer programs (though more as a gimmick), see for example here about a Commodore 64 program hidden within Prodigal’s 1984 album entitled Electric Eye.
There also was the Flexi-disc, a thin square of plastic with a program on it, which could be produced very cheap, and came as part of a computer magazine. But again, vinyl, not bakelite.