Interesting find. As far as I can tell, adding and removing tags from notes is kept in sync across platforms. But deleting a tag from the list – by right-click Delete, by Clear Unused Tags, or by Check Database – is not synced. It has to be done separately in each app – desktop, AnkiMobile, and AnkiDroid.
That’s unusual – and I haven’t found anything to suggest this is a known/expected behavior.
The sync protocol doesn’t currently have a way to say “this tag has been removed from the tag list”, so each client needs to run check database separately I’m afraid.