What you have is a hydraulic pump with a couple cylinders. Can't get much simpler than that.
You could put water in it and it would work, but probably not for long, but really you need a fluid that lubricates the working parts of the pump and keeps the seals in good condition. So really, about any hydraulic fluid that is the right viscosity would
probably work, but what Chrysler has done is spec ATF for this. It makes sense, the ATF will have lubricating properties and has additives to keep the seals in good shape and it's readily available at any repair shop or parts store.
If you really think about it, you've said you haven't had to rebuild or replace anything... and that's after 59 years or 50 years if you want to go back to 2015 when you took it apart. That sounds like they made the right decision for fluid.
Really though, ATF is pretty benign... Brake fluid, like they used to use, will eat paint where ATF won't. Yea, ATF will make a mess... So would any other fluid. But the cylinders will probably leak into the body and if you have the plastic behind the door panels, it's not going to hurt them. The pump might leak into the area under the seat and I guess could eventually drip onto the carpet, but I'll bet the top will stop working before that. I suppose it could spray onto the seat back if the conditions were right... but I've never heard of that happening and if it's like my '70 300 'vert, there's a cardboard panel that would catch the spray.
So, if it were me, I'd use ATF (I do in mine) . Any hydraulic oil is going to do the same damage if it leaked, and maybe you would see the red before a clear fluid. There's no doubt it works and no doubt it stands the test of time. IMHO, you have less chance of a leak using what Chrysler spec'd than anything you might try.
Usually there's a label on the pump. This is from a '67 Dodge. Type A superseded to Dexron and you can go down the rabbit hole of what superseded that.
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