It is a cottered crank. Crank arms themselves don't seem to be bent. It is a non-cartridge style spindle. Any ideas as to why they are not lining up?
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Like you said, they are cottered, what makes them aligned is that pin through a hole, which at the same time tightens the crank arm around the axle.
If you google "cottered crank", you can see the spindle, which has a slot in each side.
My hypothesis would be: or the cottering bold got deformed, or the spindle slot itself got deformed, or both.
A bolder hypothesis would be twisting of the spindle due to deformation or cracking, but most probably the bearings would not spin freely (how are the cranks spinning, by the way?)
The side which appears on your photo looks OK from outside, the other side we cannot tell.
But for sure this is a VERY irregular situation, which demand a full disassembly of the system, and absolutely no use before that.
Looking at the photo I notice that the nut is missing,an indication that it has been worked on at some point. With the right side pedal in the 9 o'clock position the nut side of the pin will face up verify that the leftarm pin has the nut on the bottom.Several size pins were used and it is possible that you have two different sizes.I have several cottered cranks and it seems that there is no standard as far as which direction the pin goes on which side but they are always facing in opposite directions.
One of the cotter pins is in backwards. I did that once by accident when reassembling. It's easy to fix - take the nut off, tap the pin out with a hammer, then insert it in the opposite direction from what it was before.