What qualities of bicycle bearing grease? What are the qualities of bicycle bearing grease that make it suitable for that use?

(The reason I ask is that I need to get some and I'd rather get the non-bike equivalent grease at the auto parts store a mile down the road than at the bike store 15 miles away.)

(I'm assuming the same grease can be used for bottom bracket and hub bearings -- true?)
It's a surprisingly tricky conversation. All the differences at play are so close to totally negligible, it can be easy to tally them up differently. It mostly just doesn't matter as long as it's some type of conventional bearing grease.

In terms of how well a bearing system full of a given grease is going to function, if the bike is ever going to get ridden in wet conditions, then waterproofness and washout resistance are the most important variables by far. If the bike isn't ever going to be ridden in wet conditions, then it doesn't matter at all unless you want to try to save watts with your grease. If you do then my money is on Dura-Ace grease as being the best you can get, but it's a black hole of a debate topic.

For shop use, I have come to view grease compatibility as being the most important differentiator, because it seems like getting it wrong there is the main way your choice of grease can actually have meaningful negative consequences. I'm a fan of White Lightning Crystal as general purpose grease because it's biodegradable, non-toxic and non-staining, but the problem is that aluminum based greases like it score badly on the compatibility chart, meaning arbitrarily squirting it into other people's greasy things you're adding grease to is probably a bad idea. 

Hub main bearings, bottom brackets, headsets, and pedals generally all take the same grease. Cup and cone type freehub or freewheel mechanisms generally need either heavy oil or light viscosity grease.