The Google Scholar fetcher should also fetch the DOI

No, I do not imply that JabRef discards the DOI. As far as I know Google Scholar does not include DOI field in the bibtex file it provides.

However, the answer to “The Google Scholar fetcher should also fetch the abstract” seems to insinuate that the abstract could be fetched somehow using the DOI.

If this is indeed the case then it means that you have somehow access to the DOI itself and therefore it could be also included despite google Scholar not providing it directly. If I misunderstood this point then I guess that it won’t be possible to obtain DOI from the google scholar web search option.