How would Scrum work if each Developer has his/her own unique Product Backlog?

Scrum is designed for a small team of people delivering a complex product. The roles/accountabilities, artifacts, and events are all designed to support this context. As soon as you change the context, elements of Scrum begin to break down.

Consider one example - the Daily Scrum. If you have a single Developer, what is the value of the Daily Scrum? The purpose of this event is to enable Developers to coordinate on a plan to achieve the Sprint Goal. One person has no one else to coordinate with, so you would rightfully drop this event. However, once you make such fundamental changes, what you have is no longer Scrum.

How would Scrum work for such a setup?

Put Scrum to one side for the moment, it is not implemented for its own sake. How would empiricism be established and maintained?

I suppose for one developer, Scrum involves too much overhead. Do you also have a Product Owner and a Scrum Master for each team?

Some elements will probably work, such as holding a review with stakeholders. However, if each developer does their own review, it may be too much for some stakeholders.
Some elements won't work — it's difficult to conduct a retrospective on your own.

I recommend understanding the Scrum theory and the ideas behind it, and then deciding what to do.
You could consider grouping some of the markets from your example together, with a team of developers responsible for a few of them. This way, they can learn from each other, help each other, and be much more flexible.
However, also consider whether Scrum or Agile may not be the right approach for this scenario.