they seem to give the impression that once the data goes into a GPL licensed program, it can only be extracted from that program by GPL licensed software.
This cannot possibly be correct. The GPL (and any other open source license) fundamentally derives its power from copyright law, so the absolute maximum of its reach is the same as the absolute maximum reach of copyright law: any derivative works. If something is not a derivative of the GPL'd work, then it simply isn't covered by the GPL.
The exact boundary of a derivative work is something which is jurisdiction and case dependent, but a good rule of thumb is "if the work could exist without the original, it is not a derivative work".
I am told that a plugin or addon has to be GPL.
The top voted answer on the question you are using (in a comment) as your source for your claims makes it very clear it is not as simple as this. It all depends on the degree of "entanglement" between the two components, and again comes down to whether the plugin is a derivative of the main program or not.
I have even been told that anything with a "shared data structure" counts too. So you can't access the same database either.
You are misinterpreting "shared data structure" - in this context, it means "actually using the same bytes of memory", not "uses the same schema". Again, your argument simply cannot be true because the data is (in almost every case) not a derivative work of the program that produced it, so the license of the producing program is irrelevant in copyright terms.
Some have even claimed that it goes so far as to include APIs, meaning that your program has to be GPL just to use an API of a GPL program.
You will need to provide a citation for this; put simply, people that claim this do not understand either the GPL or copyright law.