Much as Mark 1 says.
The most widely-used General Aviation hydraulic oil is known as Aeroshell Fluid 41 (dyed red). Large modern aircraft and biz jets usually have Skydrol (phosphate-ester based), which is horrible stuff if it gets in your eyes. Definitely not compatible.
I've never heard of any common piston-engined American-built light aircraft that use anything other than mineral-based oil.
Some antique aircraft may still use vegetable-based oil (natural rubber seals) but I've never come across them.
Re car brake fluid, I know of someone who filled his Cessna 210 hydraulic reservoir with it. Cost him a lot of money to have everything in the system resealed after the subsequent O ring leakage.