1👍
You have multiple options.
If you want to perform association (meaning tags and startup already exists before making the request, kinda what happen with django-admin), you could create a new serializer that have a different field for tags, accepting ids instead of the nested serializer.
If you want to have nested creation/edition, you could checkout WritableNestedSerializer from here. Because the doc says that it does not handle such usecase, because they might be many way to perform this depending on your business logic, but provide ways to perform that yourself here
Another approach would be to have a route with nested ressources (with nested routers for instance) so when you POST a tag in /startup/1/tags/ you create AND associate your tag automatically, like you did.
Now, concerning your endpoint, you need to get the data of your request and pass it to the tag serializer. This serializer will then validate your data, and if it is valid, you can perform tag creation.
To do that, you can do something like:
tag_data = request.data
tag_serializer = TagSerializer(data=request.data)
tag_serializer.is_valid()
tag = tag_serializer.save()
tag.startup_set.add(startup)
Adding the relationship have to be done in two step. You should use a transaction to ensure it is correctly created.
Also, instead of adding this logic in your view, you should override your TagSerializer/StartupSerializer create method to do that.