Oh boy, all grease is greasy but all grease is not the same. You've got your basic lithium based ngli greases and food compatible grease and marine and ultra high pressure, synthetic, non-synthetic, teflon and so forth.
If memory serves me rightt, a few years back ford had a different specification than gm and they weren't compatible. Using the wrong type often resulted in earlier failure of the greased ball or u joint.
Other than situations where the wrong grease might cause process contamination (food, chemicals, textiles, etc....) I'd say any grease is better than no grease.
Somewhere in my library I have a textbook on formulating greases for different purposes. It's about 400 pages.