So far (without expert assistance) I've learned that GREASE is basically some kind of OIL held within a "soap". The common soaps are calcium, sodium, and lithium based. The soaps can each be formulated to different hardness levels, rated by NLGI (National Lubricating Grease Institute) from 000(very soft) to 6(very hard; like a bar of, you guessed it, soap). Petroleum jelly is about 2 on that scale (but PJ is not really a grease, it's just greasy). The old, translucent Singer grease is also about 2, maybe 2.5. The degree of hardness is controlled during manufacture by using heavier or lighter "fractions" of petroleum (it's more complicated but that will suffice).
The Melt Point (temperature at which grease changes from semi-solid to liquid) is controlled by the soap base chemical. Lithium greases melt at 350-400F, sodium at 250-300F, calcium at 160-200F. PJ melts at 100-105F. The melt points can also be manipulated (within those ranges) during manufacture.
Old grease often oxidizes and gets harder, particularly if in plastic tubes, but it takes several to many years; the tin or lead tubes don't "breathe" if tightly capped and seams are undamaged but I still think it hardens somewhat given enough decades/centuries
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-Phil