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If LJ (the dev) were to make sync open source, I believe the same thing would happen to the developer of Paint.NET. Someone would just steal it and claim it as their own. That's why Paint.NET has been closed source ever since. I understand your reasoning, but the positives outweigh the negatives for me.
especially for android apps this is the reality, someone would come, copy the thing wholesale, half assedly rebrand it and then buy ads to push it on the play store, and google would do fuck all about it, it's a well documented thing
Genuine question, is it possible for a developer of a FOSS project to enforce copyright protections or does the nature of FOSS disallow that?
Yes, you are allowed to licence open source software which allows for different limitations. Common open source licences you'll see a lot of are Apache 2 which is what the Android Open Source Project uses, or GPL v2 like the Linux kernel. There are plenty of explanations online about the different licences and the limitations they impose on distribution of a piece of software.