How to make sense of a 2019 paper published in 2016 journal issue?
I would pull the paper. I would be worried that the issue is never going to be published and that the paper is going to sit in limbo for years. The fact that the journal has no web page, even in the social sciences, is concerning to me. Are you sure that libraries are going to get this newest issue when it comes out for it to be archived?

If I were to keep the paper in the pipeline for the journal, I would want some reassurances. I would look at the journal editorial board and review process. If the review process was exceptionally speedy and the initial decision was accept as is (or very minor revisions), I would worry the journal is desperate. A high rate of turn over in editors or reviewers (i.e., different people on every iteration) would also concern me. I would put a time contingency on the copyright transfer so that it expires if the issue is not published in the next few weeks.

Then you have all the issues with playing with the timeline. How are people (e.g., tenure committees or search committees) going to know when you did the work. Even saying that it was published in 2019, isn't the issue, committees want to know when you did the work. I can only imagine the headaches that it could cause with funders when trying to explain why work funded in 2018 is in a 2016 issue, regardless of when it came out. Screwing with the timeline is also probably going to mess up a lot of bibliometrics (e.g., 5 year impact factor would only go for citations through 2021 or really a 2 year window).