Shinra is derived from a Japanese yojijukugo, 神羅万象 (shinrabanshō), which is a variant of 森羅万象 (shinrabanshō).
Yojijukugo are poetic idioms that distill some sort of deep meaning into a simple phrase using only four kanji. They can be a bit difficult to translate into English. A couple easier examples: 一石二鳥 (issekinichō, "one stone, two birds") and 異体同心 (itaidōshin, "different bodies, same mind").
In this case, 神羅万象 (shinrabanshō) is, very roughly, "divine embrace, myriad phenomena." It idiomatically refers to "the entire universe," in essence, "all places, all things." I've also seen it translated as "all-covering god, ten-thousand things," or "almighty, universal."
The first kanji pair is, naturally, 神羅 (shinra). The first kanji (神, shin or kami) means "god," "almighty," or "divine." The second kanji (羅, ra) refers literally to a silk covering or net, but is actually a shortened form of 網羅 (mōra) which means "encompassing," "covering," or "embracing." Originally the phrase used 森 (shin), which means "forest," thus "forest's embrace" or "all-covering forest."
The name is essentially a reference to the company's goal of conquering the world and bending nature to their will. Potentially representative of their arrogance and hubris, as well -- playing god, as it were.
It's also worth noting that the second kanji is also used in 阿修羅 (ashura), the phonetic Japanese translation of the Sanskrit असुर (asura). Asura are a class of chaotic, power-seeking, warmongering supernatural beings in Hinduism and Buddhism. "Warring demon" is a fairly common, though not entirely accurate, translation.
It's possible that the developers also wanted to reference that, creating a dichotomy between "god" and "demon" or "divinity" and "chaos," though I find that explanation less likely. Still, it's something you can certainly read into it.