Same problem with a brand new MBP that came with 10.14.1 (Mohave release 1) installed. Both CHROME and DROPBOX have two entries in the notifications setting list in the SYSTEM PREFERENCES app. I've played with various combinations of settings across the two entries. It seems that the settings are logically OR'd -- turning a parm on in either makes that parm active. It seems likely that the problem is in the way mac os handles applications rather than something specific about the individual app. In general, all the OSes have terrible notification management -- its a maze of settings that no one could possibly keep organized. Andoird, iOS, mac OS are all incredibly badly designed in this respect. I'm somewhat surprised that no one has written an app to manage all the notifications at once -- I would pay money for that! I will cross post this into the Mohave support channel at apple.com. It is likely to be fixed in some future release. It will be interesting when that happens which of these very similar entries disappears.
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