I did a bit of research and it seems grease is grease is grease, more or less. The only real difference AFAIK between normal lithium-based car grease and the bike stuff is that the bike stuff is a little more water resistant, and the car stuff is more heat resistant. In other words, one is formulated for cooler conditions and one for hotter conditions. Automotive grease like bike grease is perfectly safe on rubber AFAIK since cars contain plenty of rubber seals as well
Apparently, however, some bike mechs use plain old lithium marine grease, which is just as water resistant as the bike stuff and a lot less money. In fact I've read a couple of comments elsewhere that imply the the fancy Phil Wood bike grease actually is just the same as plain old marine grease!
So for now I'm gonna use my dad's castrol since I cannot bear to fork out for more pricey bike grease when I have so much at home in the US.