"Roughly" is used as in "He just pushed me roughly." How did "roughly" then become used as in "They're roughly the same weight."?
Well, you're confusing yourself by starting from a usage that isn't the core meaning of "rough" at all. The core meaning of "rough" is the sense that's to do with physical texture; an irregular, broken surface as opposed to a smooth, flat one. Both meanings you're asking about arise through metaphorical analogy with the physical sense. The "pushed me roughly" derives from the concept of "rough behavior", where behavior stereotyped as associated with lower, working social classes is thought of as "rough" or "coarse", as opposed to upper classes being glorified as "smooth" and "refined". The "roughly the same" sense arises from an analogy of roughness in terms of precision and fit; think of the way two rough surfaces fit together as opposed to the way two smooth surfaces fit together.
Rough is most commonly used to refer to things that are not smooth, e.g. a rough piece of wood or rough terrain.
Whereas smooth things are perceived to be more precise, rough things are considered less carefully created or maintained. A person who is describing something that is not as precisely or finely constructed as he could like might call it a little rough around the edges.
The original meaning of rough as borrowed from the West Germanic word, rukhwaz, were similar to shaggy, uncultivated, i.e., not smooth.
With these connotations of rough, it is easy to see the progression to a different meaning: not exact, as in they're roughly the same weight.
Interestingly, the use of rough that you use, he pushed me roughly is yet another semantic descendant from the original meaning of rough. According to EtymOnline, rough having a meaning similar to rowdy was first attested in 1837. There could be several ways this meaning could have come about, but I would speculate that because a rough object doesn't feel as pleasant to feel as a smooth one, perhaps even painful in some cases, when you touch it, it is being rough to you.