The Kikkoman gluten free variant is also labelled as tamari.

Yes, and Chinese "light soy" is also similar to shoyu.

For me I always have Kikkoman in the fridge (especially because thats what wife grew up with) as the staple soy sauce. I like to dabble in having 1-2 other variants in the fridge at once, but they can tend to have too strong a flavor for some peoples taste. Or certain variants are best with certain dishes, etc.

This is the first time I hear about keeping soy sauce in the fridge. Is this common?

I keep sweet soy like kecap manis and 醬油膏 in the fridge because occasionally it can catch mold otherwise. I do the same with sweet vinegar like balsamic. However I think this depends a lot on how hot and humid your environment is. In cool and dry climates it's probably not necessary.

醬油膏 = soy sauce paste

It’s one of my favorites. I try to find ones without much additives and refrigerate after opening to keep it fresh.

Every soy sauce label I remember seeing has said, "Refrigerate after opening." I don't know why this seems to be rarely done (at least in the US).

Yeah I just follow food safety instructions on packaging

I have feeling that I should do it. The difference between open bottle that have stayed outside and fresh bottle is pretty clear. Refrigeration would slow down any reactions and thus keep taste better longer.