Given the pictures, I'd remove the interior shiplap, properly insulate and brace the wall, and then drywall, backer board, and tile. As to why....who can tell. And are you sure it's 60's era?

Why do you think the plastic didn't stop any water vapor? It doesn't need to be taped to do that. In your climate you will need a vapor retarder to avoid front in the wall cavity unless you add foam board insulation on the outside. Solve the wall design then worry about the finishes.

Also, determine how the walls are laterally braced. Is the fiber board sheathing adequate or do you need diagonal corner bracing or perhaps the interior ship lapped boards are bracing it.

Definitely early 60s home, we bought it from the original owner who also built it. I was very surprised to find this but they owned a lumber mill so perhaps it was cheaper to do this with fibreboard outside than to use plywood on the exterior for strength? Although with that mentality they probably wouldn't use plywood on so many of the interior walls either.

My major concern is that it could be structural and also interior walls are butted up against it so removal might not be an option anyway.

I guess my understanding of vapour barriers is not good enough yet, I assumed any opening would let too much moist air through but I am certainly not an expert. Anyways the only truly important thing right now is that I keep moving on this bathroom and I guess at this point leaving it as is makes the most sense. I will replace anything with water damage and move forward.

The insulation here definitely leaves something to be desired so I will look into insulating from the outside in the future.

If your climate is moderate enough, insulation board on the outside and new siding over it might eliminate the need for a vapor retarder on the inside.

The exterior sheathing is probably old Celotex "beaver board". If so, the lateral bracing of the exterior walls is probably done with diagonal "let-in" boards at the corners of the house.

It is odd that the interior wood boards are continuous behind the interior walls. Perhaps the original use was as a cabin or utility building.

I had a house built in the late 20s-early 30s that had tongue and groove walls and ceilings. The floor was kinda squishy but those walls and ceilings were incredible. I assume that house had studs but I never bothered to look for one...

Removing the shiplap just to increase the R-Value of the bathroom wall will produce no measurable improvements in energy efficiency. If it's step one in a major retrofit, feel free.

[is] there is any way
to expose the boards on some walls in the rest of the house?

Simple to do. But only advisable economically if you remove the exterior siding and insulation, then superinsulate and re-side.

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Trendy downtown Toronto 1980s renos involved removing the plaster and strapping from exterior brick walls.I can feel the wind whipping through, let alone the convection currents off the ice cold walls.

Thanks everyone for your advice and thoughts! I have done some more research on insulation and vapour barriers and we have decided to leave the shiplap in place as we think more problems will be created than solved by removing. Next year we will look at insulating over the sheathing. We may use vapour barrier paint inside depending on if the insulation will be enough to raise the temp of the sheathing enough to go without it and rely on the paper faced insulation. I don't believe the paper faced insulation is enough of a vapour barrier in our climate in the winter on its own without insulating the sheathing.

The 1980s Canadian building code mania for airtight interior poly sealed with acoustic sealant has thankfully blown over. All you need is a retarder, i.e., latex paint, and an air-tight as possible construction to avoid problems in cold and very cold climates. Arctic cold is another matter entirely!

Thanks worthy, I am planning to insulate over the sheathing next year which will hopefully contribute to a warm stud cavity with good drying capacity to the interior and exterior. From what I have read this would be a reasonable approach as long as there is the proper ratio of insulation in front of compared to behind the sheathing. Our average temps here range from +20c to -20c with some periods of -30 to -40 each winter normally.