It is already working as intended. Yes, it's weird and confusing, but that's still how it is. If you want to change this behaviour you have to take it to main meta.
Look at the last pages of the tag overview. Basically all of those are synonyms. A few are rarely used tags or even some that will be removed in some time if no questions use them anymore, but the vast majority in those last pages are tag synonyms.
You would have to go through all ten pages of narrative questions, find those that still display "storytelling" in the tag list, remove that and then... the tag would still be visible in the tag overview on the last page and will never be removed...
The reason is that synonyms on StackExchange, like in the real world, are still words that can be used. They just have the same meaning as another word in the context of "questions on Writing.StackExchange", which is why a moderator made sure they link to the same description - this is all that a "synonym" on StackExchange is. An easy way to have one tag, one short descriptive word, link to another word that desribes the same thing, ideally has a longer description and is used on many other questions. This way you can find the stuff that is basically what you are interested in ("storytelling") by searching for the term you are familiar with ("storytelling") and still find all the stuff relevant to this topic with the name that is used more often on this site ("narrative").
storytelling is here to stay. It will just link to narrative from now on.
It is still a tag. That's why it appears like a normal tag. It just has the same meaning as another tag in the context of "questions on Writing.Stackexchange", which is why it points to the other tag so that there is one central point where people interested in this topic can search.
It's also not possible to easily see from the tag overview which tags are synonyms, as some have descriptions because they were used a bit before being synonymized and some don't have a description. storytelling is a good example for this and most of the tags without a description on the last page of the tag overview.
At the bottom of the tag overview page you can find a link to the tag synonyms if you want to get a feeling for the amount of tags that will display this behaviour.
And to answer your question about whether you can use both: No. You can't use storytelling and narrative on the same question. The synonym ("storytelling") will be erased and only the base tag ("narrative") will stay.
I just tested this with the question Ways to develop characters in a fast paced video game, which I found while looking through the list with questions tagged narrative and keeping an eye open for a question that still uses storytelling. I simply added the tag narrrative and the system automatically removed storytelling. You can look at the revision history, but it shows that I replaced the tag. You can simply test it yourself if you don't believe me. There should still be a few such questions left. When posting a new question the synonym ("storytelling") will also be replaced with the base tag ("narrative") as far as I know, but I don't have a storytelling questions right now, so you need to test this yourself with a good storytelling question if you really need to know for sure.
That some questions still have both tags is because the synonmization happened after the last edit on the question. I tested this with the question you linked as you can see here. Again, the revision history shows that I removed the tag, but it happened automatically. Just check for yourself when you come across such a case at some point in the future.
But please don't flood the frontpage with retags of this synonym. Really, having a few questions tagged with the synonym is not a problem that needs to be fixed. It's a different name so that people can use the one they are more comfortable with and find the one they are searching while still getting an overview of all the stuff that is probably relevant to their search. Over time this will sort itself out anyway and our front page already has too many tag edits most of the time for the relatively low activity on Writing.SE in regards to new questions.