Will a 2015 MacBook logic board work on a 2016?
No, this will not work.
It's not the connectors you have to worry about, it's the screw hole locations and Apple is notorious for changing this very minor but critical detail to prevent this exact thing from happening.
For example, I just finished restoring a Mid-2007 (white) MacBook.  I ordered a bottom case (improperly listed as a 2007) and while every single connector lined up properly, it would not go in because of very subtle differences in where support posts and screw holes were placed - sometimes as little as a couple millimeters.

Could the pins be different?

Absolutely!  Apple makes changes all of the time depending on the components they source.  The MacBook Pro's you referenced could very well have an issue where the pins are different.  After all, it's the 2016 MacBook Pro models that suffered the &quot;flexgate&quot; (display cable) issue where the 2015's did not.  The cable was made shorter and the back light power pins (21V) were moved too close to a data pin (3V) resulting in machines that had fried GPUs.
My advise is to have the SSD repaired.  There are firms out there that will do board level repair like this for about the same price as you're going to pay for a different logic board.