I'd say it's unlikely they used different splines on each axle. But can't say for sure I've only been in three of these old truck rear ends. From my research usually the axle gears are the same for each side, but I didn't look that close other than at the 47 to 57 years (which the carrier matches my truck.) Could be your gear splines are nicked up a bit not letting it slide in... If the original axle broke and it was right at the splines very possible. If so you'll need to fill the nicks smooth... Please don't try to clean them up by pounding the axle in. You will hate yourself later.
Also are the splines keyed? Meaning they will only mesh one way,(This is usually done by half cuting the groove between two splines. I haven't seen an axle like this, but I've seen it with other things such as drive shaft yokes.
You are sure the axle is not actually entering the gear right? Because you have the other sides axle out the pinion shaft will not turn, the spiders will turn on them selves, need to watch for internal gear motion... (I'm still betting on different size axles)
The transfer case thing: Do you know how this thing shifts? I put a Transfer-case in my truck to give me a high and low range of gears. Maybe another redneck job... Could be the power tower as suggested too. Does the out put shaft speed change when it's shifted?
